Marino Morgane

Nicolas † Gourdain

Legras Guillaume

Alfano David

Jameson Schmidt Turkel

L-S Launder

Sharma Muscl

EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS)

Keywords:

The Fenestron ® has a crucial anti-torque function and its sizing is a key point of the Helicopter design, especially regarding thrust and power predictions. This paper reports the investigations done on a full scale Dauphin Fenestron ® . The objectives are first to evaluate the influence of some numerical parameters on the performance of the Fenestron ® . Then the flow is analyzed for a high incidence pitche, for which the rotor blade can experience massive boundary layer separations. Simulations are carried out on a single blade passage model. Several parameters are benched, such as grid quality, numerical schemes and turbulence modeling. A comparison with test bench measurements is carried out to evaluate the capability of the numerical simulations to predict both global performance (thrust and power) and local flows (static pressure at the shroud and radial profiles inside the vein). The analysis demonstrates the capability of numerical simulations to accurately estimate the global performance of the Fenestron ® , including at high pitch angles. However, some discrepancies remain on the local flow, especially in the vicinity of the rotor shroud. A more detailed analysis of the local flow is performed at a blade pitch angle of 35°, with a particular interest for the blade tip region.

! * ! * ! ! * ! !"# ! [ -] C ! Pressure coefficient, ( !!! ! ! !

Introduction

The Fenestron ® was developed by the engineering department of Sud-Aviation (Airbus Helicopters) in 1970's [START_REF] Mouille | Helicopter steering and propelling device[END_REF] [START_REF] Mouille | Ten years of Aerospatiale experience with the Fenestron and conventional tail rotor[END_REF] as an alternative solution to the conventional tail rotor. The principal function of the Fenestron ® is to generate a thrust to counterbalance the torque of the main rotor. Improvements, in terms of safety and noise, for light-to-medium helicopters, have made it a trademark for Airbus Helicopters. The system is composed of a shrouded rotor and topped with a large vertical fin [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF], as described on Figure 1. The shroud includes a collector with rounded lips, a cylindrical zone at the blade passage and a conical diffuser. The gearbox (hub) is supported by three arms or a stator row and fairs systems which provide power to the rotor and control the blade pitch angle. The pitch rotor monitors the rotor thrust of the Fenestron ® . In hover flight, the rotor (fan) leads the flow from the collector to the diffuser which creates the shroud effort. The thrust of the Fenestron ® is thus composed by the shroud and the rotor thrust. The sizing of the Fenestron ® is still a challenge, due to the complexity of the flow. The internal flow of the Fenestron ® is three-dimensional, turbulent and unsteady due to the interactions between fixed (shroud-stator) and rotating parts (rotor). Like in most turbomachines, secondary flows also exist in the Fenestron ® , such as the tip leakage flow induced by the clearance between the rotor and the shroud. As a consequence, the analysis of the flow in realistic industrial configurations remains very difficult, even in a wind tunnel environment. Supported by experimental campaigns [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF]- [START_REF] Morelli | Le Fenestron ® sur l'hélicoptère[END_REF], a better understanding of the flow physics in these systems can be expected thanks to recent progresses in Computational Fluid Dynamics (CFD).

Several numerical approaches have been proposed in the literature to represent the flow in Fenestron ® configurations. For example the use of a 2D-axisymmetric model has been carried out on the Dauphin helicopter in hover flight, showing good agreement with measurements [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF]. However, this approach can be used only for axial flight conditions and 3D effects, such as blade tip vortices, are not represented. Another way to model the flow in the Fenestron ® consists in solving Navier-Stokes equations on a 2D-axisymmetric shroud and by modeling the fan as a fully coupled, time averaged momentum source term in the momentum equation [START_REF] Rajagopalan | Detailed Aerodynamic Design of the RAH 66 FANTAIL USING CFD[END_REF]. While numerical predictions were in good agreement with measurements in the collector area, the pressure distribution was not accurately estimated in the blade region [START_REF] Rajagopalan | Detailed Aerodynamic Design of the RAH 66 FANTAIL USING CFD[END_REF]. Other works reported in the literature suggest to use an approach based on a 3D geometry of the shroud [START_REF] Alpman | Unsteady RAH-66 Comanche flowfield simulations[END_REF]- [START_REF] Alpman | Toward a Better Understanding of Ducted Rotor Antitorque[END_REF]- [START_REF] Gardarein | Unsteady aerodynamic and aeroacoustic simulations of a Fenestron ® tail rotor[END_REF]. The rotor is modeled by an actuator disk that uses the momentum and blade element theory to estimate the rotor thrust corresponding to a single pitch. The major benefit of the model is the computational cost reduction due to the rotor simplification. This approach allows computing the Fenestron ® performance in the whole flight domain. However, this method does not represent 3D effects in the vicinity of the blade, such as tip clearance and blade swirl. It usually results in an unrepresentative local flow into the vein [START_REF] Gardarein | Unsteady aerodynamic and aeroacoustic simulations of a Fenestron ® tail rotor[END_REF]. To improve the quality of the numerical predictions for 3D flows, the solution is thus to consider the full 3D domain with rotating (rotor) and fixed (shroud-hub-stator-fin) parts [START_REF] D'alascio | Aerodynamic study EC135 FEN HFC By means CFD[END_REF]. The advantages of this method are the accuracy of the geometry and the possibility to perform all flight conditions. Unfortunately, this process remains time consuming in terms of both mesh generation and simulation time. To balance the simulation cost, it is possible to reduce the whole domain to a single blade passage with spatial periodic conditions [START_REF] Mouterde | Aerodynamic computations of isolated Fenestron ® in hover conditions[END_REF]. This approach has been validated in hover flight on the EC135 geometry [START_REF] D'alascio | Aerodynamic study EC135 FEN HFC By means CFD[END_REF]. Despite the increase of the available computing power, only a few studies deal with the validation of the numerical method for 3D flows, as reported in [START_REF] D'alascio | Aerodynamic study EC135 FEN HFC By means CFD[END_REF]. The scope of this paper is thus to evaluate the influence of numerical parameters (mesh, scheme and turbulence) on the global performance and local flows of the Fenestron ® in hover flight.

To validate the method, numerical predictions are compared to bench test measurements on a full scale Dauphin Fenestron ® [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF]. The CFD approach is based on the single blade passage proposed by Mouterde in [START_REF] Rajagopalan | Detailed Aerodynamic Design of the RAH 66 FANTAIL USING CFD[END_REF]. Since this work focuses on the local flow and interactions between the rotor and the shroud, the stator is not modeled in order to simplify the geometry. The numerical simulation of the flow relies on a representation of the 3D flow with a Reynolds-Averaged Navier-Stokes (RANS) approach (all turbulent scales are modeled). The first part of the paper details the experimental test case and computational setup. Then the second part of the study proposes an evaluation of the meshing strategies. The Chimera technique is compared to a fully coincident meshing approach. For the chimera technique an evaluation of the meshing refinement is conducted. The third part of the paper proposes to evaluate the influence of numerical schemes and turbulence modeling on the prediction of the Fenestron ® performance.

Study case 2.1 Experimental case

The test case investigated in this paper is the Dauphin Fenestron ® investigated by Morelli and Vuillet [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF]. Experiments were carried out on a scale one test bench in hover flight. Measurements on the ducted rotor have been done to compare different rotor and shroud configurations. Among the different configurations described by Morelli and Vuillet [START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF], the reference case is chosen. It is based on a rotor with 11 equally spaced blades and a hub supported by three arms. A balance (with an accuracy of 1%) and a torque system (with an accuracy of 0.5%) are used to measure thrust and power. As illustrated in Figure 2, the local flow is evaluated by measuring the static pressure with 32 steady sensors located along the duct vein. The flow is also characterized upstream and downstream the rotor at several radial locations with a 5-hole probe.

Computational setup

The numerical simulations are performed with the code elsA, developed at ONERA [START_REF] Cambier | The Onera elsA CFD Software: Input From Research and Feedback From Industry[END_REF]. This solver is a multidisciplinary code object-oriented, specialized in both internal and external flows. This code solves both Euler and Navier Stokes equations. It is based on a cell-centered finite volume formulation on multi-block structured meshes. A wide panel of turbulence models and numerical schemes is available. The study reported in this paper relies on a steady-state RANS approach. Boundary layers are assumed to be fully turbulent due to the large Reynolds number, based on the rotor chord (Re > 10 6 ). As reported in the literature [START_REF] Denton | Some limitations of Turbomachinery CFD[END_REF], there are four main sources of errors when comparing numerical predictions with experimental data: -Accuracy of the simulated geometry (and difficulty to represent small details such as technological effects), -Boundary conditions (e.g. isothermal or adiabatic walls, far-field approximation), -Adequacy between the numerical scheme (spectral properties of the scheme) and the mesh grid quality, -Turbulence modeling (RANS, LES, etc.). This work specifically handles the two last sources of errors (discretization and turbulence modeling errors). To do this, different meshing approaches, numerical schemes and RANS-based turbulence models are benched and their influence on the global performance and local flows predictions are evaluated. For all computations, a 3D single blade passage is chosen. Periodic conditions are applied on the lateral faces of the domain. Far-field boundary conditions are used for the external boundaries of the domain. The size of the domain is 10.R in each direction (with R the radius of the Fenestron ® ) around the shroud geometry. Previous simulations have been run with a 20.R 3 box to check that the size of the domain has no influence on local flow and global performance predictions. Shroud, hub and blade walls are represented with non-slip boundary conditions. The rotor rotates around the Z-axis and the blade pitch axis is defined as the Y-axis as shown in Figure 3.

Meshing strategy for Fenestron ® issues

For structured multi-block meshes, the main difficulty lies on the grid generation process around complex geometries. Concerning the Fenestron ® , the blade pitch variation [-10°; +40°] and the rotor tip clearance represent a considerable challenge for the grid topology and the mesh quality.

First, in order to choose the correct meshing strategy, the coincident approach is compared to the chimera method. The coincident approach is the reference method to simulate with accuracy the geometry. Nevertheless, with a variation of 50° of pitch angle, a given grid topology is not adapted for all blade pitches [START_REF] D'alascio | Aerodynamic study EC135 FEN HFC By means CFD[END_REF] (so the meshing should be adapted to each blade pitch, which is not affordable to describe the whole performance curve). The second meshing strategy is the Chimera method [13] [28]. The advantage of the Chimera approach is to segregate the mesh generation of fixed and rotating parts. Nevertheless, the overset method induces conservation losses through the interpolations between meshes. Moreover fixed and rotating cannot be fully encased without an overlapped grid or a gap between them. Then, the second part of the work proposes an evaluation of the grid refinement for the chimera method. The aim is to compare four grid refinements to evaluate the influence on the performance of the Fenestron ® and on the rotor wake dissipation. All calculations presented in this section are carried out with the two-equation turbulence model k-ω of Kok [START_REF] Kok | Improvements of the Two equation Turbulence Models in Multi-Block Flow Solvers : Free-Stream Dependancy and Transition[END_REF] and a 2 nd order centered numerical scheme [START_REF] Jameson | Numerical Solution of the Euler Equations by Finite Volume Methods Using Range Kutta Time-Stepping Schemes[END_REF]. The first study is carried out on the M1 grid (see Table 1) with 6 million grid points.

Description of the coincident approach

Two strategies can be used for the meshing of the Fenestron ® with a coincident approach. On the one hand, D'Alascio et al. [START_REF] D'alascio | Aerodynamic study EC135 FEN HFC By means CFD[END_REF] generated the grid topology for high pitches and encounter difficulties for low blade pitches. On the other hand, the work reported in this paper proposes to mesh in the first instance the low pitches and to adapt the grid topology for high pitch cases. An O-grid topology is used for the shroud, hub and blade geometries. A Htopology is adopted for the blade tip area. The reference mesh is made of 156 blocks. The size of the first cell is imposed to achieve a normalized wall distance around unity (y + ~1), which is mandatory to fully resolve boundary layers. The blade grid at 50% of the span is presented on Figure 5, for a pitch of 25°.

Description of the Chimera approach

For the Fenestron ® application, the Chimera method consists in separately generating blade and shroud-hub meshes which permits to easily set up the pitch around the blade pitch axis, as shown on Figure 3. The assembly of the two meshes is realized by the application of overlapping boundary conditions and masking conditions for areas corresponding to a solid. For both boundary conditions, the transfer of conservative and turbulent variables is done through 3D non-conservative interpolations. The Alternative Digital Trees (ADT) method [START_REF] Bonet | An alternating Digital Tree (ADT) Algorithm for 3DGeometric searching and intersection Problems[END_REF] is used to solve the interpolation search cell procedure for fixed and rotating meshes. A second order interpolation is chosen to improve the accuracy of the interpolation with a particular care for grid consistence between the two meshes. An evaluation of the conservative losses inherent to the Chimera technique is proposed in Section 4.3. An encased problem is induced by the Chimera approach, which results in a non-physical root gap of a size similar to the shroud gap. The grid topology for the rotor blade is based on a C-H topology around the blade profile, as presented in Figure 4. Blade extremities (tip and root) are discretized by a H fluid volume topology. The first cell is set in order to obtain y + =1 at wall. The whole blade mesh contains 22 blocks, representing 1.5x10 6 cells. The background mesh includes the shroud and the hub geometries. The grid generation is first done in 2D and is then extruded in a periodic 3D mesh. An O-block is chosen for the meshing of shroud and hub walls. The size of the first cell is also set to y + =1. The whole background mesh is composed of 21 blocks and 4.5x10 6 cells.

Comparison of the two strategies

This part is dedicated to the comparison between the coincident approach and the Chimera technique. It is first proposed to evaluate the conservative losses of the Chimera method, as it is performed for turbomachinery [START_REF] Castillon | Numerical simulations of technological effects encountered on turbomachinery configurations with the Chimera technique[END_REF]. To quantify the losses induced by the overset method, the mass flow difference ΔQ is estimated from plane 1 (upstream the rotor) to plane 3 (downstream the rotor), as: with Q the mass flow. The mass flow difference is close to 1.0% for a grid of 6x10 6 cells (grid M1). With the finest grid (grid M3, 23x10 6 cells) the mass flow loss reduces to 0.1%. Figure 6 compares numerical predictions with the two strategies to measurements. The global thrust as well as the shroud and the rotor thrusts are presented. According to the Froude theory [START_REF] Leishman | Principles of HELICOPTER AERODYNAMICS[END_REF], for both approaches, the total thrust of the system is the result of half rotor and shroud thrust (as also reported in wind tunnel campaigns [START_REF] Morelli | Le Fenestron ® sur l'hélicoptère[END_REF]). Until a blade rotor pitch of +30°, the numerical predictions are not modified by the grid strategy. Then, from +30° to +40°, the coincident approach overestimates the Fenestron ® thrust compared to measurements. This is related to an overestimation of the rotor thrust contribution. For analyzing the influence of the grid strategy on the local flow, normalized static pressure on the shroud for blade pitch +35° is presented on Figure 7. In the collector area, the static pressure distributions are equivalent for both strategies. It is coherent with the shroud polar curves observed in Figure 6. With the coincident

Q = Q p1 Q p3 Q p1
approach, the static pressure distribution is affected by the mesh discretization, in the vicinity of the blade (Figure 7b). As a consequence, periodic blade wakes are quicker dissipated compared to the Chimera approach, due to the low-quality grid in the region of the shroud.

For the Chimera approach, Figure 8 points out the dissipation of the rotor wake at the trailing edge due to the interpolation inherent to the method. With the coincident approach, the wake is not dissipated but the resolution of the flow around the leading edge is influenced by the mesh quality.

To conclude, the Chimera approach is better adapted than the coincident approach, to describe the flow in a large range of blade pitches. To overcome this limitation, the coincident grid should be redesigned for each blade pitch, which is a costly task if automatic adaptive grid methods are not available. At this step, the use of the Chimera approach is thus preferable.

Analysis of grid refinement

The effect of grid refinement is carried out for 4 different grids, from 4x10 6 cells (grid M0) to 23x106 cells (grid M3), as shown in Table 1. All computations in this section are performed with the two-equation turbulence model of Kok [START_REF] Kok | Improvements of the Two equation Turbulence Models in Multi-Block Flow Solvers : Free-Stream Dependancy and Transition[END_REF] with the SST correction and a 2nd order centered scheme [START_REF] Jameson | Numerical Solution of the Euler Equations by Finite Volume Methods Using Range Kutta Time-Stepping Schemes[END_REF]. Global performance predictions A comparison of the numerical prediction with the bench test is carried out on Figure 9 a) and b). The higher the pitch, the more the performance prediction is sensitive to grid refinement. A coarse mesh overestimates the thrust at blade pitches higher than +35 o . A better grid refinement reduces the difference with measurements.

Mesh

Local performance predictions

The effect of the refinement is analyzed for high blade pitch (+35°). Figure 9 c) presents the pressure coefficient distribution through the vein (from the collector lip to the diffuser lip). The points A to D correspond to the shroud, as presented in Figure 2. From point A to B, the upper collector zone evolves in a low-pressure area, where the suction peak is reached at the maximal curvature radius. This region of the shroud generates most of the thrust. Then, in the blade region, from point B to C, the decrease of the pressure is related to the presence of the blade tip vortices. This phenomenon will be better described in part 5. Through the diffuser, from point C to point D, the pressure returns to the ambient static pressure value. For all mesh refinements, numerical results and tests follow the same trend. The pressure on the collector is more sensitive to grid refinement than the rest of the flow. Only refinements above grid M1 are in good agreements with measurements. The numerical simulations with grids M2 and M3 predict a stagnation zone at the end of the collector and before the blade passage. On the diffuser area, the solution is sensitive to the junction of the blade with the diffuser.

The grid refinement has also an influence on the pressure peak. Figure 9 d) shows the radial profile of the axial velocity, extracted from plane 2, as a function of the radial position in the duct. Blade velocity distributions are obtained using an azimuthal mass flow averaging. Three zones are underlined. The first zone, from h/H=0 to h/H=0.05, corresponds to the blade root region which is influenced by the non-physical root leakage flow. The second one is the "linear zone" of the rotor, from h/H=0.05 to h/H=0.9. The third area highlights the effect of blade tip vortices, for h/H>0.9. In the linear zone, numerical predictions follow the same trend as the measurements. The axial velocity is overestimated at both the shroud and the root. In the blade tip region, axial velocity is over predicted by 20% for the grid M0 to 30 % for the grid M3. This observation is correlated with the suction peak that is discussed in part 5.

Effect on the rotor wake

Figure 10 presents the axial velocity at h/H=0.85 for the M1 and M3 grids at pitch +35° (the solution of the blade grid is overlapped on the background grid), in order to show the wake dissipation. As expected, the wake dissipation is reduced with the finest grid (M3). This observation is in agreement with conservatives losses described in section 4.3. Therefore to preserve the wake of the rotor blade, the radial refinement of the grid should by three times more important than with a coincident approach. To complete the grid refinement study, the influence of the numerical scheme is estimated. Three classical numerical schemes are tested. The second order centered scheme of Jameson Schmidt Turkel (JST) [START_REF] Jameson | Numerical Solution of the Euler Equations by Finite Volume Methods Using Range Kutta Time-Stepping Schemes[END_REF] is compared with the 2 nd order Advection Upstream Splitting Method (AUSMP+ (P)) [START_REF] Liou | 30 the computational fluid Dynamics, chap AUSMp schemes and extensions for low mach and multiphase[END_REF] and a 3 rd order scheme of Roe [START_REF] Roe | Approximate Riemann Solvers Parameters Vectors and Differences Schemes[END_REF]. To stabilize the JST scheme, an artificial viscosity term is added with a scalar artificial viscosity. The linear fourth order dissipation term k4 is set to 0.016. All simulations presented in this section are conducted on grid M1 with the two-equation turbulence model of Kok [START_REF] Kok | Improvements of the Two equation Turbulence Models in Multi-Block Flow Solvers : Free-Stream Dependancy and Transition[END_REF]. Figure 11 e) shows the convergence of the numerical solution for the three schemes. The residual decrease by three orders of magnitude, which is sufficient to consider that the convergence is achieved.

Global performance predictions

The numerical power prediction is compared to experimental data on Figure 11 a). For a high level of thrust, the increase of the scheme order improves the power estimation. However, on the thrust polar, Figure 11 b), the third order scheme of Roe overestimates the thrust compared to experimental data. For high pitch level, a balance is established between the accuracy of the power estimation, which needs a low dissipation from the numerical schemes, and the smoothing of the flow oscillations to reach the correct rate of thrust.

Local performance predictions

Figure 11 c) presents the pressure coefficient as a function of the vein distance for a blade rotor pitch of +35°, at the shroud. On the collector area, the pressure distribution is very sensitive to the numerical scheme. The JST scheme is in good agreement with the experimental bench data while the 3 rd order scheme overestimates the static pressure coefficient at this location. Through the blade zone (from B to C) and in the vicinity of the diffusor, the solution is not sensitive to the numerical scheme. Figure 11 d) presents the axial velocity distribution on plane 2 for a blade rotor pitch of +35°, in the vein. The solution is not sensitive to the numerical schemes in the linear zone and in the vicinity of the blade tip. Indeed, the over-prediction of the velocity deficit in the tip region should not be attributed to the grid/scheme combination. Actually, most discrepancies are observed in the blade root region, as shown in Figure 11 e) at plane 3.

A qualitative study of the blade tip region has been conducted for blade rotor pitch +35°. The Q-criterion [START_REF] Jeong | On the identification of a Vortex[END_REF] is taken positive to point out the blade tip vortices, Figure 12. The convection of the tip vortex is influenced by the numerical scheme (the 3 rd order scheme transports the tip vortex further downstream than 2 nd order schemes). The interaction between the blade vortex flow and the shroud is affected by the numerical scheme. The 3 rd order scheme predicts the development of small flow patterns that are not observed with 2 nd order schemes. This mechanism is related to the creation of a "mirror vortex" at the shroud, as described by Cerra and Smith [START_REF] Cerra | Experimental observations of vortex ring interaction with the fluid adjacent to a surface[END_REF]. As a consequence, the wall induces the generation of a secondary vortex, rotating in the opposite direction. The mechanism is well shown in Figure 12, especially with the 3 rd order scheme.

Actually, the sensitivity of the solution to the numerical scheme, especially the predictions of the power, increases with the blade pitch angle.

Influence of turbulence modeling

The turbulence modeling is a key issue in RANS-based simulations. Different first order turbulence models are compared in this section: the one transport equation model of Spalart-Allmaras S-A [START_REF] Spalart | One-Equation Turbulence Model for Aerodynamic Flows[END_REF] is compared with the two transport equations model of k-ε (Launder Sharma L-S) [START_REF] Launder | Application of the Energy Dissipation Model of Turbulence to the Calculation of Flows near a Spinning Disk[END_REF] and k-ω (Kok [START_REF] Kok | Improvements of the Two equation Turbulence Models in Multi-Block Flow Solvers : Free-Stream Dependancy and Transition[END_REF] and Menter [START_REF] Menter | Improved two equation K-ω Turbulence Models for aerodynamic flows[END_REF]) with Shear Stress Transport (SST) correction. The SST correction proposed in [START_REF] Menter | Zonal two equation k-omega Turbulence models for aerodynamic flows[END_REF] avoids the delay in the prediction of adverse pressure gradient effects. This correction should thus be helpful for the flow prediction at high pitch angles, for which the adverse pressure gradient is more important. All simulations presented in this section are performed on grid M1 with the 2 nd order centered JST scheme. Figure 13 f) shows the convergence of the numerical solution with the four turbulence models. The residual drops by 3 or 4 orders of magnitude.

Global performance predictions

Figure 13 a) presents the power estimation depending on the turbulence modeling. In comparison with the experimental data, the thrust-power polars predicted by the Menter and Kok models overestimate at a high level of thrust. The S-A model reveals to be extremely dissipative. For the L-S model, a stall region appears. The thrust polar, shown on Figure 13 b), highlights two zones in comparison with experimental measurements. Two zones are observed: a first zone which is linear corresponds to low pitch angles, from [-10°; +35°] and a second zone which correspond to high pitch angles from [+35°; +40°]. In the linear zone, the two transport equation models (L-S, Kok, Menter) overestimate the thrust predictions compared to the S-A model. Differences increase at high blade rotor pitch. The L-S model predicts a region of stalled flow earlier compared to other models.

Local performance predictions

An analysis of the local flow is carried out on the blade rotor pitch angle of +35°. Pressure coefficient distributions at the shroud are presented on Figure 13 c). From point A to point B, at the inlet lip, turbulence models have an important influence on the pressure peak predictions, as shown in Table 2 A secondary increase of the pressure coefficient is observed with the L-S model, close to point B, which does not appear in the experimental measurements and with other turbulence models. However, all turbulence models show an increase of the pressure coefficient between point B and point C, related to the breakdown of the tip clearance vortex.

Figure 13 d) shows the radial profiles of normalized axial velocity at plane 2. The magnitude of the axial velocity is overestimated by most of turbulence models (except the L-S. k-ε model) compared to measurements, in the tip region (by about 20% with S-A. and k-ω models). This observation underlines the difficulty for first order turbulence models to accurately describe turbulence in the vicinity of the tip clearance. In the linear zone, h/H=0.05 to h/H=0.9, all turbulence models predict the same trend. At the blade root, k-ε and k-ω models are more influenced by the root leakage flow than other turbulence models. The S-A predicts a higher level of turbulent kinetic energy than other turbulence models, resulting in an increase of the turbulent viscosity. This behavior limits the nonphysical root tip flow, which helps to match the measurements (however not for the good reasons).

Figure 13 e) shows the radial profiles of axial velocity distribution at plane 3. The differences between turbulence models predictions are more important at this section than at plane 2. The main reason is the influence of the size of the stalled flow that is predicted at plane 2, which impose a redistribution of the axial velocity inside the vein: if the mass flow is reduced at the shroud, it will increase the mass flow at the shroud. As for the prediction of the flow in the tip region at plane 2, it shows the difficulty to accurately predict the flow in such a configuration when separation occurs at high blade pitches.

This section shows that the global performance of the Fenestron ® is sensitive to both numerical scheme and turbulence modeling. The power prediction is more influenced by the numerical schema while the thrust is more influenced by the turbulence modeling. For local performance, the turbulence modeling is of paramount importance in the vicinity of the blade tip clearance.

Comments on the blade region

In this section, the k-ω model of Kok and the 2 nd centered order scheme of Jameson are used. The finest grid M3 is chosen (23x10 6 cells) to conduct the local flow analysis in the vicinity of the blade. Most discrepancies between numerical predictions and measurements appear in the blade passage (from point B to point C). In this region two flow mechanisms interact together. First, the blade tip vortex is generated by the tip clearance, close to the shroud. Then the boundary layer of the shroud interacts with this vortex, leading to a secondary flow. This shear flow affects the tip region performance by promoting a three-dimensional separation [START_REF] Peacock | Blade tip gap effects in Turbomachines-A Review[END_REF]. Figure 15 presents a 3D view of the pressure coefficient, showing the effect of the blade tip vortex. The boundary layer of the shroud is sucked by the blade tip vortex therefore a suction peak appears in the coefficient of pressure (Figure 16). Such a suction peak has already been reported in the literature [START_REF] Rajagopalan | Detailed Aerodynamic Design of the RAH 66 FANTAIL USING CFD[END_REF]. Such 3D flows are known to be challenging both for experimental sensors and numerical simulations, so it can explain, at least partially, the discrepancies between numerical simulations and measurements. However, since this part of the shroud is parallel to the streamwise flow, it does not affect the shroud thrust.

In order to highlight the difficulty to simulate the flow in the blade region, two pitches around the blade pitch angle of +35° are studied. "eps" is a small variation of the blade pitch around pitch +35° (about a degree). Figure 16 presents the distribution of the pressure coefficient for the three different pitches. The pressure coefficient distribution is well captured at the three pitch angles compared to measurements. Thereafter, the behavior of the suction peak in the blade region depends on the blade pitch angle. The higher the pitch angle, the less the suction peak in the blade region. Moreover, the pressure coefficient before the blade passage and after the collector depends also on the blade pitch angle.

Figure 17 presents the turbulent kinetic energy along the cylindrical part of the shroud. Three zones are plotted; zone 1 and zone 3 are extracted at 6% of blade chord before and after the blade, respectively, and zone 2 is extracted at the middle of the blade position. As expected, a peak of turbulent kinetic energy is observed at zone 2, which is related to the tip clearance flow. However, the zone 1 reveals also a peak of turbulent kinetic energy (upstream the blade passage) due to the shroud boundary layer.

Figure 18 shows the flow field colored with the axial velocity, for the three blade pitch angles in the cylindrical part. Correlated with the pressure coefficient (Figure 15), three zones are highlighted: 1) a large velocity area in the collector region (which participated to the shroud thrust), 2) a backflow that develops upstream the blade for blade pitch angle of +35°-eps and +35° and 3) the tip clearance flow. The behavior of the backflow zone is sensitive to the blade pitch angle. Above a blade pitch angle of +35°, the axial velocity imposed by the rotor becomes sufficient to delay the separation zone upstream the blade, Figure 18 c).

Conclusion

An evaluation of the influence of several CFD numerical parameters has been done to evaluate their influence on the aerodynamic performance and the local flow predictions of a Fenestron ® configuration. Regarding the large variation of the pitch angle in such a configuration, [-10 o . +40 o ], the grid design remains an issue. Two methods have been compared: a classical coincident approach (a grid is designed for each blade pitch) and an overlapping approach (Chimera method). While the coincident method avoids the use of interpolations between grids of different density, it reaches quality limits for high pitch angles (an over-dissipation is observed at such flow conditions due to the stretching of the mesh). In that regards, the use of the Chimera approach helps to overcome this difficulty.

If a sufficient care is brought to the grid, most numerical parameters do not influence the prediction of the Fenestron ® performance until the blade pitch +35 o . Beyond this pitch, discrepancies appeared. On the pressure distribution, the prediction of the collector area is influenced by the grid refinement and the numerical scheme. For high pitch angles, the turbulence modeling has also a major effect on the flow prediction. This is related to the difficulty for a RANS-based approach to tackle with complex flow phenomena such as boundary layers separations. Some discrepancies are observed between numerical predictions and measurements, especially in the vicinity of the tip region and at high pitch angles (where flow separations are observed). A dependence of the flow solution to a small variation of the blade pitch angle exists at such high blade pitches. A separated zone upstream the blade is observed for blade pitch angle until +35 o . Nevertheless, this work has shown that these errors cannot be attributed to the grid quality or the numerical scheme. The choice for the turbulence model has a large influence on the flow prediction. As a perspective to this work, unsteady RANS and Large Eddy Simulation will be carry out on this configuration to better understand the role of flow unsteadiness and turbulence, and provide data about turbulence properties in the vicinity of the tip clearance.

Figure 1 :

 1 Figure 1: overview of the Fenestron ® principle

Figure 2 :

 2 Figure 2: Locations of static pressure measurements at the shroud (from Morelli and Vuillet [3])

Figure 3 :

 3 Figure 3 : Chimera grid assembly

Figure 4 :Figure 5 :

 45 Figure 4: 2D view of the rotor grid with the Chimera approach at h/H=0.85

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Comparison of the thrust coefficient between Chimera technique, the coincident approach and measurements

Figure 9 :

 9 Figure9: Numerical comparison of grid refinement with experimental data base[START_REF] Morelli | New Aerodynamic Design of the Fenestron ® for Improved Performance[END_REF]

Figure 10 :

 10 Figure 10: Axial velocity iso-contours with h/H=0.85

Figure 11 :Figure 12 :

 1112 Figure 11: Numerical comparison of convective scheme for reference mesh

Figure 13 :Figure 14 :

 1314 Figure 13: Numerical comparison of turbulence modeling for reference mesh

Figure 15 :Figure 16 :Figure 17 :Figure 18 :

 15161718 Figure 15: Distribution of the coefficient of pressure on the Fenestron ® for grid M3 -Pitch +35°

Table 1 :

 1 Mesh grid refinement

		M0 (coarse)	M1 (reference)	M2 (intermediate)	M3 (fine)
	Boundary layer (Normal	73	109	205	313
	direction)
	Vein (Z-axial direction)	82	123	255	395
	Total mesh size (millions of	4	6	12	23
	points)

Table 2 :

 2 . The best agreement with measurements is observed with the k-ω model of Menter (error is 2% with respect to measurements). Comparison of turbulence models with measurements for the prediction of the pressure peak coefficient, for a blade rotor pitch angle of +35°

	Error in %	Spalart-Allmaras	k-ω Menter	k-ω Kok	k-ε Launder Sharma
	-Cp	14	2	6	6

c) Pitch +35°+eps


Luigi Santocanale

Alan Turing Cs

Kurt Gödel

Alan Turing

Relational lattices via duality

Keywords:

 observed that these two operations are the meet and join in a class of lattices-called the relational lattices-and proposed lattice theory as an alternative algebraic approach to databases. Aiming at query optimization, Litak et al.

. The contributions of this paper are as follows. Let A be a set of column's names and D be a set of cell values; we characterize the dual space of the relational lattice R(D, A) by means of a generalized ultrametric space, whose elements are the functions from A to D, with the P (A)-valued distance being the Hamming one but lifted to subsets of A. We use the dual space to present an equational axiomatization of these lattices that reflects the combinatorial properties of these generalized ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that these equations correspond to combinatorial properties of the dual spaces of lattices, in a technical sense analogous of correspondence theory in modal logic. In particular, this leads to an exact characterization of the finite lattices satisfying these equations.

Introduction

Tropashko [START_REF] Tropashko | Relational algebra as non-distributive lattice[END_REF] has recently observed that the natural join and the inner union, two fundamental operations of the relational algebra initiated by Codd [START_REF] Codd | A relational model of data for large shared data banks[END_REF]-the algebra by which we construct queries-can be considered as the meet and join operations in a class of lattices, known by now as the class of relational lattices. Elements of the relational lattice R(D, A) are the relations whose variables are listed by a subset of a total set A of attributes, and whose tuples' entries are taken from a set D. Roughly speaking, we can consider a relation as a table of a database, its variables as the columns' names, its tuples being the rows.

Let us illustrate these operations with examples. The natural join takes two tables and constructs a new one whose columns are indexed by the union of the headers, and whose rows are the glueings of the rows along identical values in common columns. As we emphasize in this paper the lattice theoretic aspects of the natural join operation, we shall depart from the standard practice of denoting it by the symbol and use instead the meet symbol ∧.

The inner union restricts two tables to the common columns and lists all the possible rows. The following example suggests how to construct, using this operation, a table of users given two (or more) tables of people having different roles.

Considering the lattice signature as a subsignature of the relational algebra, Litak, Mikulás and Hidders [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] proposed to study the equational theory of the relational lattices. The capability to recognize when two queries are equivalentthat is, a solution to the word problem of such a theory-is of course an important step towards query optimization. Spight and Tropashko [START_REF] Spight | Relational lattice axioms[END_REF] exhibited equational principles in a signature strictly larger than the one of lattice theory. A main contribution of Litak et al. [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF]-a work to which we are indebted in many respects-was to show that the quasiequational theory of relational lattices with the header constant is undecidable. The authors also proposed a base of equations for the theory in the signature extended with the header constant, and exhibited two non-trivial pure lattice equations holding on relational lattices. It was argued there that the lattice R(D, A) arises via a closure operator on the powerset P (A D A ) and, at the same time, as the Grothendieck construction for the functor P (D (-) ), from P (A) op to SL ∨ (the category of complete lattices and join-preserving mappings), sending X ⊆ A to D X and then D X covariantly to P (D X ).

The focus of this paper is on the pure lattice signature. We tackle the study of the equational theory of relational lattices in a coalgebraic fashion, that is, by using the duality theory developed in [START_REF] Santocanale | A duality for finite lattices[END_REF] for finite lattices and here partially extended to infinite lattices. Let us recall some key ideas from the theory, which in turn relies on Nation's representation Theorem [14, §2]. For a complete lattice L, a join-cover of x ∈ L is a subset Y ⊆ L such that x ≤ Y . A lattice is pluperfect if it is a complete spatial lattice and every join-cover of a completely join-irreducible element refines to a minimal one-see Section 2 for a complete definition. Every finite lattice is pluperfect; moreover, relational lattices are pluperfect, even when they are infinite. This property, i.e. pluperfectness, allows to define the dual structure of a lattice L, named the OD-graph in [START_REF] Nation | An approach to lattice varieties of finite height[END_REF]. This is the triple J (L), ≤, m with J (L) the set of completely join-irreducible elements, ≤ the restriction of the order to J (L), and the relation j m C holds when j ∈ J (L), C ⊆ J (L), and C is a minimal join-cover of j. The original lattice L can be recovered up to isomorphism from its OD-graph as the lattice of closed downsets of J (L)-where a downset X ⊆ J (L) is closed if j m C ⊆ X implies j ∈ X.

We characterize the OD-graph of the lattice R(D, A) as follows. Firstly recall from [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] that we can identify completely join-irreducible elements of R(D, A) with elements of the disjoint sum A D A . The order on completely join-irreducible elements is trivial, i.e. it is the equality. All the elements of A are join-prime, whence the only minimal join-cover of some a ∈ A is the singleton {a}. The minimal join-covers of elements in D A are described via an ultrametric distance valued in the join-semilattice P (A); this is, morally, the Hamming distance, δ(f, g) = {x ∈ A | f (x) = g(x)}. Whenever f, g ∈ D A we have f m δ(f, g) ∪ {g} and these are all the minimal join-covers of f .

As in correspondence theory for modal logic, the combinatorial structure of the dual spaces is an important source for discovering axioms/equations that uniformly hold in a class of models. For relational lattices, most of these combinatorial properties stem from the structure of the ultrametric space (D A , δ). When we firstly attempted to show that equations (RL1) and (RL2) from [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] hold in relational lattices using duality, we realized that the properties necessary to enforce these equations were the following: P1. Every non-trivial minimal join-cover contains at most one join-irreducible element which is not join-prime.

Moreover, the generalized ultrametric space

(D A , δ) is P2. symmetric, i.e. δ(f, g) = δ(g, f ), for each f, g ∈ D A , P3. pairwise complete: if δ(f, g) ⊆ X ∪ Y , then δ(f, h) ⊆ X and δ(h, g) ⊆ Y for some h ∈ D A .
Various notions of completeness for generalized ultrametric spaces are discussed in [START_REF] Ackerman | Completeness in generalized ultrametric spaces. p-Adic Numbers Ultrametric Anal[END_REF]. At first we called pairwise completeness the Beck-Chevalley-Malcev property of (D A , δ). Indeed, it is equivalent to saying that the functor P (D (-) ) : P (A) op --→ SL ∨ mentioned above sends a pullback square (i.e., a square of inclusions with objects X∩Y, X, Y, Z) to a square satisfying the Beck-Chevalley condition. As the property implies that a collection of congruences of join-semilattices commute, it is also a sort of Malcev condition. We show with Theorem 4 that property P1 of an OD-graph is definable by an equation that we name (Unjp). We investigate the deductive strength of this equation and show in particular that (RL2) is derivable from (Unjp), but not the converse.

In presence of P1, symmetry and pairwise completeness can also be understood as properties of an OD-graph. Symmetry is the following property: if

k 0 m C ∪ {k 1 } with k 1 not join-prime, then k 1 m C ∪ {k 0 }. Pairwise complete- ness can be read as follows: if k 0 m C 0 ∪ C 1 ∪ {k 2 } with k 2 not join-prime and C 0 , C 1 , {k 2 } pairwise disjoint, then k 0 m C 0 ∪ {k 1 } and k 1 m C 1 ∪ {k 2 } for some completely join-irreducible element k 1 .
We exhibit in Section 6 three equations valid on relational lattices and characterize, via a set of properties of their OD-graphs, the pluperfect lattices satisfying (Unjp) and these equations. We propose these four equations as an axiomatization of the theory of relational lattices that we call [[AxRel]]. The main result of this paper, Theorem 7, sounds as follows. If we restrict to finite lattices that are atomistic-that is, lattices in which any element is the join of the atoms below it, so the order on join-irreducible elements in the dual space is trivial-then a lattice satisfies [[AxRel]] if and only if its OD-graph is symmetric and pairwise complete, in the sense just explained.

We can build lattices similar to the relational lattices from P (A)-valued ultrametric spaces. It is tempting to look for further equations so to represent the OD-graph of finite atomistic lattices satisfying these equations as P (A)-valued ultrametric space. Unfortunately this is not possible, since a key property of the OD-graph of lattices of ultrametric spaces-the ones ensuring that the distance function is well defined-is not definable by lattice equations. Yet Theorem 7 also exhibits a deep connection between the OD-graph of finite atomistic lattices satisfying [[AxRel]] and the frames of the commutator logic [S5] A , see [START_REF] Kurucz | Combining modal logics[END_REF]. Considering the complexity of the theory of combination of modal logics, Theorem 7 can be used to foresee and shape future researches. For example, we shall discuss in Section 7 how to derive undecidability results from the correspondent ones in multidimensional modal logic. In particular, a refinement of the main result of Litak et al. [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF]Corollary 4.8] can be derived.

The paper is structured as follows. We introduce in Section 2 the notation as well as the least lattice theoretic tools that shall allow the reader to go through the paper. In Section 3 we describe the relational lattices, present some known results in the literature, and give a personal twist to these results. In particular, we shall introduce semidirect products of lattices, ultrametric spaces as a tool for studying relational lattices, emphasize the role of the Beck-Chevalley property in the theory. In Section 4 we characterize the OD-graphs of relational lattices. In Section 5 we present our results on the equation (Unjp). In Section 6, we describe our results relating equations valid on relational lattices to symmetry and pairwise completeness. In the last Section we discuss the results presented as well as ongoing researches, by the author and by other researchers, trace a road-map for future work.

Some elementary lattice theory

A lattice is a poset L such that every finite non-empty subset X ⊆ L admits a smallest upper bound X and a greatest lower bound X. We assume a minimal knowledge of lattice theory-otherwise, we invite the reader to consult a standard monograph on the subject, such as [START_REF] Davey | Introduction to Lattices and Order[END_REF] or [START_REF] Grätzer | General Lattice Theory[END_REF]. The technical tools that we use may be found in the monograph [START_REF] Freese | Free lattices[END_REF], that we also invite to explore. A lattice can also be understood as a structure A for the functional signature (∨, ∧), such that the interpretations of these two binary function symbols both give A the structure of an idempotent commutative semigroup, the two semigroup structures being tied up by the absorption laws x∧(y∨x) = x and x∨(y∧x) = x. Once a lattice is presented as such structure, the order is recovered by stating that x ≤ y holds if and only if x ∧ y = x.

A lattice L is complete if any subset X ⊆ L admits a smallest upper bound X. It can be shown that this condition implies that any subset X ⊆ L admits a greatest lower bound X. A complete lattice is bounded, since ⊥ := ∅ and := ∅ are respectively the least and greatest elements of the lattice.

A closure operator on a complete lattice L is an order-preserving function j : L --→ L such that x ≤ j(x) and j 2 (x) = j(x), for each x ∈ L. We shall use Clop(L) to denote the poset of closure operators on L, under the pointwise ordering. It can be shown that Clop(L) is itself a complete lattice. If j ∈ Clop(L), then the set L/j of fixed points of j is itself a complete lattice, with L/j X = L X and L/j X = j( L X). For the correspondence between closure operators and congruences in the category of complete join-semilattices, see [START_REF] Joyal | An extension of the Galois theory of Grothendieck[END_REF].

Let L be a complete lattice. An element j ∈ L is said to be completely joinirreducible if j = X implies j ∈ X, for each X ⊆ L; the set of completely join-irreducible element of L is denoted here J (L). A complete lattice is spatial if every element is the join of the completely join-irreducible elements below it. An element j ∈ J (L) is said to be join-prime if j ≤ X implies j ≤ x for some x ∈ X, for each finite subset X of L. We say that j ∈ J (L) is non-join-prime if it is not join-prime. An atom of a lattice L is an element of L such that ⊥ is the only element strictly below it. A spatial lattice is atomistic if every element of J (L) is an atom.

For j ∈ J (L), a join-cover of j is a subset X ⊆ L such that j ≤ X. For X, Y ⊆ L, we say that X refines Y , and write X Y , if for all x ∈ X there exists y ∈ Y such that x ≤ y. A join-cover X of j is said to be minimal if j ≤ Y and Y X implies X ⊆ Y ; we write j m X if X is a minimal join-cover of j. In a spatial lattice, if j m X, then X ⊆ J (L). If j m X, then we say that X is a non-trivial minimal join-cover of j if X = {j}. It is common to use the word perfect for a lattice which is both spatial and dually spatial. We need here something different: Definition 1. A complete lattice is pluperfect if it is spatial and for each j ∈ J (L) and X ⊆ L, if j ≤ X, then Y X for some Y such that j m Y . The OD-graph of a pluperfect lattice L is the structure J (L), ≤, m .

That is, in a pluperfect lattice every cover refines to a minimal one. Notice that every finite lattice is pluperfect. If L is a pluperfect lattice, then we say that X ⊆ J (L) is closed if it is a downset and j m C ⊆ X implies j ∈ X. As from standard theory, the mapping X → { Y ⊆ J (L) | X ⊆ Y, Y is closed} defines a closure operator whose fixed points are exactly the closed subsets of J (L). The interest of considering pluperfect lattices stems from the following representation Theorem.

Theorem 1 (Nation [START_REF] Nation | An approach to lattice varieties of finite height[END_REF]). Let L be a pluperfect lattice and let L(J (L), ≤, m ) be the lattice of closed subsets of J (L). The mapping

l → {j ∈ J (L) | j ≤ l} is a lattice isomorphism from L to L(J (L), ≤, m ).
It was shown in [START_REF] Santocanale | A duality for finite lattices[END_REF] how to extend this representation theorem to a duality between the category of finite lattices and the category of OD-graphs. The following Lemma shall be repeatedly used in the proofs of our statements.

Lemma 1. Let L be a pluperfect lattice, let j m C and k ∈ C. If j ≤ D with D { (C \ {k}), k}, then k ∈ D. In particular, if k < k, then { C \ {k}, k } is not a cover of j.

The relational lattices R(D, A)

In this Section we define relational lattices, recall some known facts, and develop then some tools to be used later, semidirect products of lattices, generalized ultrametric spaces, a precise connection to the theory of combination of modal logics (as well as multidimensional modal logic and relational algebras).

Let A be a collection of attributes (or column names) and let D be a set of cell values. A relation (or, more informally, a table) on A and D is a pair (X, T ) where X ⊆ A and T ⊆ D X ; X is the header of the table while T is the collection of rows. Elements of the relational lattice R(D, A) are relations on A and D.

Before we define the natural join, the inner union operations, and the order on R(D, A), let us recall a few key operations.

If X ⊆ Y ⊆ A and f ∈ D Y , then we shall use f X ∈ D X for the restriction of f to X; if T ⊆ D Y , then T X shall denote projection to X, that is, the direct image of T along restriction, T X := {f X | f ∈ T }; if T ⊆ D X , then i Y (T ) shall denote cylindrification to Y , that is, the inverse image of restriction, i Y (T ) := {f ∈ D Y | f X ∈ T }. Recall that i Y is right adjoint to X .
With this in mind, the natural join and the inner union of tables are respectively described by the following formulas:

(X 1 , T 1 ) ∧ (X 2 , T 2 ) := (X 1 ∪ X 2 , T ) where T = {f | f X i ∈ T i , i = 1, 2} = i X1∪X2 (T 1 ) ∩ i X1∪X2 (T 2 ) , (X 1 , T 1 ) ∨ (X 2 , T 2 ) := (X 1 ∩ X 2 , T ) where T = {f | ∃i ∈ {1, 2}, ∃g ∈ T i s.t. g X 1 ∩X 2 = f } = T 1 X1∩X2 ∪ T 2 X1∩X2 .
The order is then given by

(X 1 , T 1 ) ≤ (X 2 , T 2 ) iff X 2 ⊆ X 1 and T 1 X2 ⊆ T 2 .
It was observed in [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] that R(D, A) arises-as a category with at most one arrow between two objects-via the Grothendieck construction for the functor sending X ⊆ A contravariantly to D X and then D X covariantly to P (D X ). Let us record the following important property: Lemma 2. The image of a pullback square by the functor P (D (-) ) : P (A) op --→ SL ∨ satisfies the Beck-Chevalley property.

P (D X1∩X2 ) i X 2 + + P (D X2 ) X 1 ∩X 2 o o P (D X1 ) i X 3 + + X 1 ∩X 2 O O P (D X3 ) X 2 O O X 1 o o
The above statement means that if we apply the functor to inclusions of the form X 1 ∩X 2 ⊆ X i ⊆ X 3 , i = 1, 2, then the two possible diagonals in the diagram on the right, X2 • i X3 and i X2 • X1∩X2 , are equal. The Beck-Chevalley property is a consequence of the glueing property of functions:

if f ∈ D X2 , g ∈ D X1 and f X 1 ∩X 2 = g X 1 ∩X 2 , then there exists h ∈ D X3 such that h X 2 = f and h X 1 = g.
We can recast the previous category-theoretic observations in an algebraic framework. An action of a complete lattice L over a complete lattice M is a monotonic mapping : L --→ Clop(M ), thus sending X ∈ L to a closure operator X on M . Given such an action, if we define j(X, T ) := (X, X T ), then j(X, T ) is a closure operator on the product L × M . In particular, the set of j-fixed points,

L j M := {(X, T ) ∈ L × M | X T = T }, is itself a complete lattice,
where the meet coincides with the one from L×M , while the join is given by the formula (X

1 , T 1 )∨ L j M (X 2 , T 2 ) := (X 1 ∨X 2 , X 1 ∨X 2 (T 1 ∨T 2 )
). We call L j M the semidirect product of L and M via j. The naming is chosen here after the semidirect product of groups, which is a similar instance of the Grothendieck construction. Given such an action, the correspondence X → M/ X gives rise to a covariant functor from L to the category SL ∨ , so that it makes sense to ask when the Beck-Chevalley property holds, as in Lemma 2. This happens-and then we say that an action satisfies the Beck-Chevalley property-exactly when

X 1 ∨ X 2 T = X 1 X 2 T , for each X 1 , X 2 ∈ L and T ∈ M . (1)
Notice that the identity X 1 X 2 T = X 2 X 1 T is a consequence of (1). As these closure operators correspond to congruences of complete join-semilattices, we also think of the Beck-Chevalley property as a form of Malcev property, stating that a collection of congruences (thought as binary relations) pairwise commute (w.r.t composition of relations).

Relational lattices from ultrametric spaces. Let us come back to the lattice R(D, A). Define on the set D A the following P (A)-valued ultrametric distance:

δ(f, g) := {x ∈ A | f (x) = g(x)} .
Thus δ(f, f ) ⊆ ∅ and δ(f, g) ⊆ δ(f, h) ∪ δ(h, g) for any f, g, h ∈ D A , making (D A , δ) into a generalized metric space in the sense of [START_REF] Lawvere | Metric spaces, generalized logic and closed categories[END_REF]. 1 With respect to the latter work-where axioms for the distance are those of a category enriched over (P (A) op , ∅, ∪)-for f, g ∈ D A we also have that δ(f, g) = ∅ implies f = g and symmetry, δ(f, g) = δ(g, f ). We can define then an action of P (A) on P (D A ):

X T = {f ∈ D A | ∃g ∈ T s.t. δ(f, g) ⊆ X} . (2)
We can now restate (and refine) Lemma 2.1 from [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF]-which constructs the lattice R(D, A) via a closure operator on P (A + D A )-as follows:

Theorem 2. The correspondence sending (X, T ) to (A \ X, i A (T )) is an isomorphism bewteen the relational lattice R(D, A) and P (A) j P (D A ).

The action defined in (2) satisfies the identity (1). As a matter of fact, ( 1) is equivalent to pairwise completeness of (D A , δ) as an ultrametric space, see [START_REF] Ackerman | Completeness in generalized ultrametric spaces. p-Adic Numbers Ultrametric Anal[END_REF], namely the following property:

if δ(f, g) ⊆ X 1 ∪ X 2 ,
then there exists h such that δ(f, h) ⊆ X 1 and δ(h, g) ⊆ X 2 .

(

) 3
It is easily verified that ( 3) is yet another spelling of the glueing property of functions.

Observe that, given any generalized ultrametric space (F, δ) whose distance takes values in P (A), equation ( 2)-with D A replaced by F -defines an action of P (A) on P (F ). The lattice P (A) j P (F ) shall have similar properties to those of the lattices R(D, A) and will be useful when studying the variety generated by the relational lattices. As an example, we construct typed relational lattices, i.e. lattices of relations where each column has a fixed type. To this goal, fix a surjective mapping π : D --→ A. For each a ∈ A, we think of the set D a = π -1 (a) as the type of the attribute a. Let S(π) be the set of sections of π, that is, s ∈ S(π) if and only if s(a) ∈ D a , for each a ∈ A. Notice that (S(π), δ) is a pairwise complete sub-metric space of (D A , δ). The lattice R(π) := P (A) j P (S(π)) is the typed relational lattice. It can be shown that relational lattices and typed relational lattices generate the same variety.

Relational lattices from multidimensional modal logics. In order to illustrate and stress the value of identity (1), i.e. of the Beck-Chevalley-Malcev property, we derive next a useful formula for computing the join of two tables under the P (A) j P (D A ) representation.

(X 1 , T 1 ) ∨ (X 2 , T 2 ) = (X 1 ∪ X 2 , X 1 ∪ X 2 (T 1 ∪ T 2 )) = (X 1 ∪ X 2 , X 1 ∪ X 2 T 1 ∪ X 1 ∪ X 2 T 2 ) since the modal operators X are normal, in the usual sense of modal logic, = (X 1 ∪ X 2 , X 2 X 1 T 1 ∪ X 1 X 2 T 2 ) by (1) = (X 1 ∪ X 2 , X 2 T 1 ∪ X 1 T 2 ) since X 1 T 1 = T 1 and X 2 T 2 = T 2 .
Theorem 2 suggests that a possible way to study the equational theory of the lattices R(D, A) is to interpret the lattice operations in a two sorted modal logic, where the modal operators are indexed by the first sort and act on the second. It is easily recongnized that each modal operator satisfies the S5 axioms, while equation (1) implies that, when A is finite, each modal operator X is determined by the modal operators of the form a with a an atom below X.

That is, the kind of modal logic we need to interpret the lattice theory is the commutator logic

[S5] n = [S5, . . . , S5 n-times
], with n = card A, see [START_REF] Kurucz | Combining modal logics[END_REF]Definition 18].

Minimal join-covers in R(D, A)

The lattices R(D, A) are pluperfect, even when A or D is an infinite set. The completely join-irreducible elements were characterized in [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] together with the meet-irreducible elements and the canonical context (see [START_REF] Davey | Introduction to Lattices and Order[END_REF]Chapter 3] for the definition of canonical context). If we stick to the representation given in Theorem 2, the completely join-irreducible elements are of the form a = ({a}, ∅) and f = (∅, {f }). We can think of a as an empty named column, while f is an everywhere defined row. They are all atoms, so that, in particular, we shall not be concerned with the restriction of the order to J (R(D, A)) (since this order coincides with the equality). In order to characterize the OD-graph of the lattice R(D, A), we only need to characterize the minimal join-covers. Taking into account that if an element j ∈ J (L) is join-prime, then it has just one minimal join-cover, the singleton {j}, the following Theorem achieves this goal.

Theorem 3. The lattices R(D, A) are atomistic pluperfect lattices. As a matter of fact, every element a, a ∈ A, is join-prime; for f ∈ D A , the minimal joincovers of f are of the form

f ≤ a∈δ(f,g) a ∨ g , for g ∈ D A .
The proof of this statement is almost straightforward, given the characterization of R(D, A) as the semidirect product P (A) j P (D A ) and the definition of the closure operators given with equation [START_REF] Codd | A relational model of data for large shared data banks[END_REF]. For this reason, we skip it.

In particular, every minimal join-cover contains at most one non-join-prime element. In view of Theorem 1, we obtain a more precise description of the closure operator described in [12, Lemma 2.1] that gives rise to relational lattices.

Corollary 1. The relational lattice R(D, A) is isomorphic to the lattice of closed subsets of A D A , where a subset X is closed if δ(f, g)∪{g} ⊆ X implies f ∈ X.

In order to ease the reading, we shall use in the rest of this paper the same notation for a completely join-irreducible element of R(D, A) and an element of A D A . This is consistent with the above Corollary, as under the isomorphism we have a = {a} and f = {f }. Thus a shall stand for a, and f for f .

In a spatial lattice L (thus in a relational lattice), an inequation s ≤ t holds if and only if j ≤ s implies j ≤ t, for each j ∈ J (L)-thus we shall often consider inequations of the form j ≤ s with j ∈ J (L). When L = R(D, A), the characterization of minimal join-cover leads to the following principle that we shall repeatedly use:

f ≤ x 1 ∨ . . . ∨ x n iff δ(f, g) ∪ {g} {x 1 , . . . , x n }, for some g ∈ D A . ( 4
)

Uniqueness of non-join-prime elements

For an inclusion we mean a pair (s, t) of terms (in the signature of lattice theory) such that the equation t ∨ s = s (i.e. the inequality t ≤ s) is derivable from the usual axioms of lattices. Thus, the equality s = t reduces to the inequality s ≤ t.

We write s ≤ t for a lattice inclusion and say it holds in a lattice if the identity s = t holds in that lattice. Next, let us set

d (u) := u 0 ∧ (u 1 ∨ u 2 ) , d ρ (u) := (u 0 ∧ u 1 ) ∨ (u 0 ∧ u 2 ) , so d (u) ≤ d ρ (u)
is (an inclusion equivalent to) the usual distributive law. Consider the following inclusion:

x ∧ (d (y) ∨ d (z) ∨ w) (Unjp) ≤ (x ∧ (d ρ (y) ∨ d (z) ∨ w)) ∨ (x ∧ (d (y) ∨ d ρ (z) ∨ w)) .
Theorem 4. The inclusion (Unjp) holds on relational lattices. As a matter of fact, (Unjp) holds in a pluperfect lattice if and only if every minimal join-cover contains at most one non-join-prime element.

Proof. Let us prove the first statement. To this goal, it will be enough to argue that any join-irreducible element below the left-hand side of the inclusion is below its right-hand side. Let k be such a join-irreducible element. It is not difficult to see that if k is join-prime, then k is also below the right-hand side of the inclusion. Suppose then that k is non-join-prime, whence k = f for some f ∈ D A . From f ≤ d (y) ∨ d (z) ∨ w and (4), it follows that there exists g ∈ D A such that δ(f, g) ∪ {g} {d (y), d (z), w}. In particular, {g} {d (y), w} or {g} {d (z), w}. We firstly suppose that the last case holds. If a ∈ δ(f, g) and a ≤ d (y) = y 0 ∧ (y 1 ∨ y 2 ), then a ≤ (y 0 ∧ y 1 ) ∨ (y 0 ∧ y 2 ) = d ρ (y), since a is join-prime. It follows that δ(f, g) ∪ {g} {d ρ (y), d (z), w}, whence f ≤ x ∧ (d (y) ∨ d (z) ∨ w). If {g} {d (y), w}, then we conclude similarly that f ≤ x ∧ (d (y) ∨ d ρ (z) ∨ w). Whence k is below the right-hand side of this inclusion, and the inclusion holds since k was arbitrary.

We leave the reader to generalize the argument above so to prove that if a pluperfect lattice is such that every minimal join-cover has at most one nonjoin-prime element, then (Unjp) holds. For the converse we argue as follows.

Let L be a pluperfect lattice, let k x ∈ J (L), C x ⊆ J (L) with k x m C x , and suppose that k y , k z ∈ C x are distinct and non-join-prime. For u ∈ {y, z}, since k u is non-join-prime, there is a non-trivial minimal join-cover k u m C u ; as every non-trivial minimal join-cover has at least two elements, let C u,1 , C u,2 be a partition of C u such that C u,i = ∅ for each i = 1, 2.

We construct a valuation which fails (Unjp). Let x := k x , y 0 := k y , z 0 := k z , w := (C x \ {k y , k z }) and, for u ∈ {y, z} and i = 1, 2, let u i := C u,i . The lefthand side of the (Unjp) evaluates to k x . Assume, by the way of contradiction, that (Unjp) holds, so k x is below the right-hand side of the inclusion. Since the only minimal join-cover

D of k x such that D {k x } is {k x }, either k x ≤ d ρ (y)∨ d (z) ∨ w or k x ≤ d (y) ∨ d ρ (z) ∨ w; let
us assume that the first case holds. We have then k x ≤ d ρ (y)∨k z ∨ (C x \{k y , k z }) = d ρ (y)∨ (C x \{k y }). Considering that d ρ (y) ≤ k y , Lemma 1 implies that k y = d ρ (y) = (y 0 ∧ y 1 ) ∨ (y 0 ∧ y 2 ). Since k y is join-irreducible k y = y 0 ∧ y i for some i ∈ {1, 2}. Yet this is not possible, as such relation implies that C y,i is a join-cover of k y ; considering that C y,i is a proper subset of the minimal join-cover C y , this contradicts the minimality of C y . If k x ≤ d (y) ∨ d ρ (z) ∨ w, then we get to a similar contradiction. Whence, k x is not below the right-hand side of (Unjp), which therefore fails.

It is worth noticing that the statement "every minimal join-cover contains exactly one non-join-prime element" is not definable by equations: for A = D = {0, 1}, there is a sublattice of R(D, A) which fails this property.

While Theorem 4 gives a semantic characterization of (Unjp), we might also wish to measure its power at the syntactic level. Theorem 5 and Corollary 2 illustrate the deductive strength of (Unjp), by pinpointing an infinite set of its consequences.

Theorem 5. If s = s ρ and t = t ρ are equations valid on distributive lattices, then the equation

(x ∧ (s ∨ t ∨ w)) ∨ (x ∧ (s ρ ∨ t ρ ∨ w)) = (x ∧ (s ρ ∨ t ∨ w)) ∨ (x ∧ (s ∨ t ρ ∨ w))
is derivable from (Unjp) and general lattice axioms.

Proof. For a lattice term s, let dnf(s) be its disjunctive normal form. Recall that we can obtain dnf(s) from s by means of a sequence s = s 0 , . . . , s n = dnf(s) where, for each i = 0, . . . , n -1, s i+1 is obtained from s i by one application of the distributive law at the toplevel of the term, and by general lattice axioms. Thus, for two lattice terms s 1 , s 2 , let s 1 i , i = 0, . . . , n, and s 2 j , j = 0, . . . , m be the sequences leading to the respective normal forms.

For i = 0, . . . , n and j = 0, . . . , m, let now t i,j = x ∧ (s 1 i ∨ s 2 j ∨ w). Using (Unjp) and general lattice axioms, we can compute as follows:

t 0,0 = t 1,0 ∨ t 0,1 = t 2,0 ∨ t 1,1 ∨ t 0,2 = . . . = j=0,...,m t n,j ∨ i=0,...,n t i,m ? = t n,0 ∨ t 0,m ,
where only the last equality needs to be justified. Notice that the relation s k i+1 ≤ s k i holds in every lattice. Whence we have s k i ≤ s k i when i < i , and both t n,j ≤ t n,0 and t i,m ≤ t i,0 . It follows that the indexed join at the last line evaluates to t n,0 ∨ t 0,m . We have derived, up to now, the identity

x ∧ (s 1 ∨ s 2 ∨ w) = (x ∧ (dnf(s 1 ) ∨ s 2 ∨ w)) ∨ (x ∧ (s 1 ∨ dnf(s 2 ) ∨ w))
for every pair of lattice terms s 1 and s 2 .

Let us call co-clause a conjunction of variables. By using lattice axioms only, we can suppose that, within dnf(t), there are no repeated literals in co-clauses and that no co-clause subsumes another. Under this assumption, we have that an identity s = s ρ holds in all distributive lattices if and only if dnf(s ) is equal to dnf(s ρ ). Whence, to derive the statement of the Theorem, we can compute as follows:

(x ∧ (s ∨ t ρ ∨ w)) ∨ (x ∧ (s ρ ∨ t ∨ w)) = (x ∧ (dnf(s ) ∨ t ρ ∨ w)) ∨ (x ∧ (s ∨ dnf(t ρ ) ∨ w)) ∨ (x ∧ (dnf(s ρ ) ∨ t ∨ w)) ∨ (x ∧ (s ρ ∨ dnf(t ) ∨ w)) = (x ∧ (dnf(s ρ ) ∨ t ρ ) ∨ w) ∨ (x ∧ (s ∨ dnf(t ) ∨ w)) ∨ (x ∧ (dnf(s ) ∨ t ∨ w)) ∨ (x ∧ (s ρ ∨ dnf(t ρ ) ∨ w)) = (x ∧ (dnf(s ) ∨ t ∨ w)) ∨ (x ∧ (s ∨ dnf(t ) ∨ w)) ∨ (x ∧ (dnf(s ρ ) ∨ t ρ ∨ w)) ∨ (x ∧ (s ρ ∨ dnf(t ρ ) ∨ w))
-where we have permuted the order of the four joinands

= (x ∧ (s ∨ t ∨ w)) ∨ (x ∧ (s ρ ∨ t ρ ∨ w)) .
In [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] two equations were shown to hold on relational lattices. One of them is (RL2) that we describe next. Set

d o (u) := (u 0 ∨ u 1 ) ∧ (u 0 ∨ u 2 ) , d o ρ (u) := u 0 ∨ (u 1 ∧ u 2 ) , the equation is x ∧ (d o (y) ∨ d o (z)) ≤ (x ∧ (d o ρ (y) ∨ d o (z))) ∨ (x ∧ (d o (y) ∨ d o ρ (z))) (RL2)
Corollary 2. If s = s ρ and t = t ρ are equations valid on distributive lattices, then the inequation

(x ∧ (s ∨ t ∨ w)) ≤ (x ∧ (s ρ ∨ t ∨ w)) ∨ (x ∧ (s ∨ t ρ ∨ w)) (5)
is derivable from (Unjp). In particular (RL2) is derivable from (Unjp).

The Corollary follows from the Theorem and from the fact that x ≤ x ∨ y. In order to derive (RL2) from ( 5) (if we do not include the bottom constant ⊥ as part of the signature of lattice theory), we instantiate s := d o (y), s ρ := d o ρ (y), t := d o (z), t ρ := d o ρ (z), and w := d o ρ (z). It can be shown that (Unjp) is not derivable from (RL2)-mainly due to the role of the variable w in the (Unjp). The construction of a lattice L satisfying (RL2) but failing (Unjp) proceeds via the construction of its OD-graph J(L), ≤, m . Due to the consistent number of variables in the two equations, an automated tool such as Mace4 [START_REF] Mccune | Prover9 and Mace4[END_REF] could not help finding a countermodel. Similarly, automated tools such as Prover9 and Waldmeister [START_REF] Mccune | Prover9 and Mace4[END_REF]6] were of no help to show that (RL2) is a consequence of (Unjp).

Natural questions-e.g. decidability-may be raised concerning the equational theory of (Unjp). Since we can give an easy semantic proof that an equation of the form (5) holds on finite lattices (or pluperfect) satisfying (Unjp), a reasonable conjecture is that this theory has some sort of finite model property. Yet, proving this might not be immediate, since the variety of lattices satisfying (Unjp) is not locally finite (i.e., not every finitely generated lattice satisfying (Unjp) is finite). The construction used in [START_REF] Santocanale | A duality for finite lattices[END_REF]Proposition 7.5] may be used to argue that the lattice freely generated in this variety by three generators is infinite.

Due to its syntactic shape (Unjp) falls in a class of inclusions described in [START_REF] Santocanale | A duality for finite lattices[END_REF]Section 8] that admit a correspondent property in the OD-graph. Here, the meaning of the word correspondent is analogous to its use in modal logic, where some formulas might be uniformly valid in a frame if and only if the frame satisfies a correspondent first order property. Thus Theorem 4 is not completely unexpected. A more surprising result comes from considering the three equations below, that fall outside the syntactic fragment described in [START_REF] Santocanale | A duality for finite lattices[END_REF]; a strengthening of Lemma 1 (Lemma 4 to follow) allows to characterize the OD-graphs of pluperfect lattices satisfying (Unjp) and these equations, see Theorem 8.

x ∧ (y ∨ z) ≤ (SymPC)

(x ∧ (y ∨ (z ∧ (x ∨ y)))) ∨ (x ∧ (z ∨ (y ∧ (x ∨ z)))) x ∧ ((y ∧ z) ∨ (y ∧ x) ∨ (z ∧ x)) ≤ (x ∧ y) ∨ (x ∧ z) (VarRL1) x ∧ ((x ∧ y) ∨ d (z)) ≤ (x ∧ ((x ∧ y) ∨ d ρ (z))) ∨ (x ∧ d (z)) (RMod)
Let us first illustrate the way in which these equations hold in relational lattices.

In particular, the proof shall illustrate the crucial role played by symmetry and pairwise completeness-i.e., condition (3)-of the ultrametric space (D A , δ).

Theorem 6. The inclusions (SymPC), (VarRL1), (RMod) hold in relational lattices.

Proof. (SymPC). Let k be a join-irreducible element below x∧(y ∨z). If k is joinprime, then k is also below (x∧y)∨(x∧z), whence it is below the right-hand side of this inclusion. Therefore, let k be non-join-prime, so k = f for some f ∈ D A ; by (4), let g ∈ D A be such that δ(f, g) ∪ {g} {y, z}. Let us suppose first that g ≤ z. Since δ(f, g) {y, z}, using pairwise completeness we can find h such that δ(f, h) {y} and δ(h, g) {z}. It follows that h ≤ δ(h, g) ∨ g ≤ z; moreover, since f ≤ x, δ(h, f ) = δ(f, h), and δ(f, h) {y}, then h ≤ δ(h, f ) ∨ f ≤ x ∨ y. Consequently, we have h ≤ z ∧ (x ∨ y) and, considering that δ(f, h) {y}, we have f ≤ x ∧ (y ∨ (z ∧ (x ∨ y))). If g ≤ y, then we similarly deduce that f ≤ x ∧ (z ∨ (y ∧ (x ∨ z))). In both cases, f is below the right-hand side of this inclusion.

(VarRL1). Let k be below the left-hand side of this inclusion. If k is joinprime, then it is below the right-hand side of this inclusion as well. Otherwise k = f is non-join-prime and δ(f, g) ∪ {g} {y ∧ z, y ∧ x, z ∧ x} for some g ∈ D A . Since g ≤ r for some r ∈ {y ∧ z, y ∧ x, z ∧ x}, we consider three cases; by pairwise completeness we can also assume that g is the only element of δ(f, g) ∪ {g} below r-since if δ(f, g ) {y ∧ z, y ∧ x, z ∧ x} \ {r} and δ(g , g) {r}, then g ≤ r. Also, the last two cases, g ≤ y ∧ x and g ≤ z ∧ x, are symmetric in y and z, so that we consider among them the second-to-last only.

Suppose firstly that g ≤ y ∧z. Then, from δ(g, f ) = δ(f, g) {x∧y, x∧z} {x} and f ≤ x, we deduce g ≤ x; whence g ≤ x ∧ y and f ≤ (x ∧ z) ∨ (x ∧ y).

Suppose next that g ≤ x ∧ y. By pairwise completeness, let h be such that δ(f, h) {x∧z} and δ(h, g) {y∧z}. We deduce then h ≤ y from δ(h, g) {y} and g ≤ y, and h ≤ x, from δ(h, f ) = δ(f, h) {x} and f ≤ x. Thus h ≤ x ∧ y and f ≤ (x ∧ z) ∨ (x ∧ y).

(RMod). Let k be a join-irreducible below the left-hand side of this inclusion. If k is join-prime, then k is below x ∧ ((x ∧ y) ∨ d ρ (z)). Otherwise k = f and, for some g ∈ D A , δ(f, g) ∪ {g} {x ∧ y, d (z)}. If g ≤ x ∧ y, then all the elements that are not below x ∧ y are below d (z) and join-prime, whence they are below d ρ (z). It follows that f ≤ x ∧ ((x ∧ y) ∨ d ρ (z)). Otherwise g ≤ d (z) and, by pairwise completeness, we can also assume that g is the only element below d (z), so δ(f, g) {x ∧ y}. It follows then that δ(g, f

) ∪ {f } = δ(f, g) ∪ {f } {x}, g ≤ x, whence g ≤ x ∧ d (z). Consequently, f ≤ (x ∧ y) ∨ (x ∧ d (z)) ≤ (x ∧ ((x ∧ y) ∨ d ρ (z))) ∨ (x ∧ d (z)).
In [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] a second inclusion was shown to hold on relational lattices:

x ∧ ((y ∧ (z ∨ x)) ∨ (z ∧ (y ∨ x))) ≤ (x ∧ y) ∨ (x ∧ z) (RL1)
The same kind of tools used in the proof of Theorem 6 can be used to argue that this inclusion holds on relational lattices. The reader will have noticed the similarity of (VarRL1) with (RL1). As a matter of fact, (VarRL1) was suggested when trying to derive (RL1) from (Unjp) and the other equations as in the following Proposition.

Proposition 1. (RL1) is a consequence of (Unjp), (RMod) and (VarRL1).

Proof. Using (Unjp) and considering that y ∧ z ≤ z ∧ (y ∨ x), we have:

x ∧ ((y ∧ (z ∨ x)) ∨ (z ∧ (y ∨ x))) = (x ∧ ((y ∧ x) ∨ (z ∧ (y ∨ x)))) ∨ (x ∧ ((y ∧ (z ∨ x)) ∨ (z ∧ x)))) .
Using now (RMod) and considering that x ∧ z ≤ y ∨ x, we compute as follows:

x ∧ ((y ∧ x) ∨ (z ∧ (y ∨ x))) = (x ∧ ((y ∧ x) ∨ (z ∧ y) ∨ (z ∧ x))) ∨ (x ∧ z ∧ (y ∨ x)) = (x ∧ ((y ∧ x) ∨ (z ∧ y) ∨ (z ∧ x))) ∨ (x ∧ z) = x ∧ ((y ∧ z) ∨ (y ∧ x) ∨ (z ∧ x)) .
Considering the symmetric role of y and z, we obtain:

x ∧ ((y ∧ (z ∨ x)) ∨ (z ∧ (y ∨ x))) = x ∧ ((y ∧ z) ∨ (y ∧ x) ∨ (z ∧ x)) = (x ∧ y) ∨ (x ∧ z) , by (VarRL1).
We present now what we consider our strongest result in the study of the equational theory of relational lattices. To this end, let us denote by [[AxRel]] the (set composed of the) four equations (Unjp), (VarRL1), (RMod) and (SymPC). Also, given that we restrict to lattices satisfying (Unjp), and considering the characterization given with Theorem 4, it is convenient to introduce the notation

k 0 C m k 1 for the statement k 0 , k 1 ∈ J(L), k 1 is non-join-prime, k 1 ∈ C, and k 0 m C ∪ {k 1 }.
Theorem 7. Let L be a finite atomistic lattice. Then L |= [[AxRel]] if and only if every nontrivial minimal join-cover contains exactly one non-join-prime element and, moreover, the following properties hold in the OD-graph:

• If k 0 C m k 1 , then k 1 C m k 0 . (6)
• If k 0 C0 C1 m k 2 , then k 0 C0 m k 1 and k 1 C1 m k 2 , for some k 1 ∈ J (L).
Given Theorem 7, it becomes tempting to look for a representation Theorem. Given a pluperfect atomistic lattice satisfying the above four equations, we would like to define an ultrametric space on the set of non-join-prime elements with distance valued on the powerset of the join-prime ones, and then argue that the lattice constructed via the standard action, defined in [START_REF] Codd | A relational model of data for large shared data banks[END_REF], is isomorphic to the given lattice. Unfortunately, this idea does not work, since if we try to set

δ(k 0 , k 1 ) = C whenever k 0 C m k 1 , this might be ill defined since the implication "k 0 C m k 1 and k 0 D m k 1 implies C = D" might fail.
Moreover, there is no equation nor quasiequation enforcing this, as an immediate consequence of the next Proposition. Proposition 2. There is an atomistic sublattice of R({0, 1}, {0, 1}) which does not arise from an ultrametric space.

Theorem 7 is a consequence of a more general Theorem, to be stated next, characterizing the OD-graphs of pluperfect lattices in the variety axiomatized by [[AxRel]]. While the conditions stated next may appear quite complex, they are the ones to retain if we aim at studying further the theories of relational lattices by duality-e.g., a sublattice of a relational lattice need not be atomistic. Theorem 8. A pluperfect lattice belongs to the variety axiomatized by [[AxRel]] if and only if every minimal join-cover contains at most one non-join-prime element and, moreover, the following properties hold in its OD-graph:

• If k 0 m C, then there exists at most one c ∈ C with c ≤ k 0 .

(π-VarRL1)

• If k 0 C m k 1 , then no element of C is below k 0 . (π-RMod) • if k m C 0 C 1 with C 0 , C 1 non-empty, then for some k ∈ J (L), either k m {k } C 1 , k m C 0 , and k ≤ C 1 ∨ k, or k m C 0 {k }, k m C 1 , and k ≤ C 0 ∨ k. (π-StrongSymPC)
Let us notice that the conditions

• If k 0 C0 C1 m k 2 then, for some k 1 ∈ J (L), k 0 C0 m k 1 , k 1 C1 m k 2 , and k 1 ≤ C 0 ∨ k 0 . (π-SymPC) • If k 0 C m k 1 , then k 1 ≤ C ∨ k 0 (π-Sym)
follow from the above properties. On atomistic pluperfect lattices the last condition is equivalent to (6).

Lemma 3. If L is a pluperfect lattice with L |= (Unjp) and whose OD-graph satisfies (π-StrongSymPC) and (π-RMod), then (π-SymPC) holds as well.

Proof. Let k 1 m C 0 C 1 {k 2 } with k 2 ∈ J (L) and non-join-prime, and use (π-StrongSymPC) to find k 1 such that either (i)

k 0 m {k 1 } C 1 {k 2 } and k 1 ≤ k 0 ∨ C 1 ∨ k 2 , or (ii) k 0 m C 0 {k 1 } and k 1 ≤ k 0 ∨ C 0 .
Let us argue, by contradiction, that (i) cannot arise. By (Unjp), k 1 is join-prime, whence the relation

k 1 ≤ k 0 ∨ C 1 yields k 1 ≤ k 0 .
This, however, contradicts (π-RMod).

We close this section by proving Theorem 8. To this end, we need a generalization of Lemma 1. As the refinement relation is an extension to subsets of the order relation, the relation m , defined next, can be considered as an extension to subsets of the minimal join-covering relation. Lemma 4. Let L be a pluperfect lattice and let j m C 0 C 1 . Suppose that X ≤ C 0 and j ≤ X ∨ C 1 . Then there exists a minimal join-cover of the form j m D 0 C 1 with D 0 X and D 0 m C 0 .

While the proof that a pluperfect lattice whose OD-graphs satisfies those properties essentially mimics the proof of Theorem 6, we prove instead the converse direction through a series of Lemmas. whence the left-hand side of (VarRL1) evaluates to k 0 , which is therefore below the right-hand side of this inclusion. It follows that either k 0 ≤ y, or k 0 ≤ z, in both cases contradicting the fact that C is a minimal join-cover.

The inclusion

x ∧ (y ∨ d (z)) ≤ (x ∧ (y ∨ d ρ (z))) ∨ (x ∧ (y ∨ (d (z) ∧ (y ∨ x))) , (Sym)
is derivable from (Unjp), (SymPC) and (RMod). It can also be shown that (RMod) is a consequence of (Sym).

Lemma 6. If (Sym) holds in a pluperfect lattice, then its OD-graph satisfies (π-Sym).

Proof. Suppose that the inclusion holds and let k 0 C m k 1 . Since k 1 is non-joinprime, there exists a minimal join-cover k 1 m D which we can partition into two non empty subsets D 1 and D 2 . Let now x := k 0 , y := C, z 0 := k 1 , z 1 = D 1 , z 2 = D 2 . Then, the left-hand side of the inclusion evaluates to k 0 , which therefore is below the right-hand side. Considering that x is k 0 and that the unique minimal join-cover of k 0 whose elements are all below k 0 is {k 0 }, it follows that either k 0 ≤ y ∨ d ρ (z) or k 0 ≤ y ∨ (d (z) ∧ (y ∨ x)).

Argue that d ρ (z) < d (z), since k 1 is join-irreducible, whence by Lemma 1, {y, d ρ (z)} is not a cover of k 0 , excluding the first case. Therefore {y, d (z) ∧ (y ∨ x)} is a cover of k 0 , whence, by Lemma 1, k 1 = d (z) ∧ (y ∨ x), showing that k 1 ≤ y ∨ x = C ∨ k 0 and proving the statement.

k 1 ≤ C 1 ∨ C 0 ∨ k 0 .
We consider next equation (RMod). Put

x := k 0 , y := C 0 , z 0 := C 1 ∨ k 1 , z 1 := C 1 ∨ C 0 , z 2 := k 0 . From k 1 ≤ C 1 ∨ C 0 ∨k 0 , we get z 0 ∧(z 1 ∨z 2 ) = z 0 .
Whence, the left-hand side of (RMod) evaluates to k 0 so k 0 is below the righthand side of (RMod). Considering that {k 0 } is the unique minimal join-cover of k 0 whose elements are all below k 0 , it follows that either k

0 ≤ z 0 ∧ (z 1 ∨ z 2 ) or k 0 ≤ y ∨ (z 0 ∧ z 1 ) ∨ (z 0 ∧ z 2 ). As k 0 ≤ C 1 ∨ k 1 = z 0 , it follows that k 0 ≤ y ∨ (z 0 ∧ z 1 ) ∨ (z 0 ∧ z 2 )
. We can use then Lemma 4 to deduce that k 0 has a minimal join-cover of the form

k 0 m C 0 D, with D {z 0 ∧ z 1 , z 0 ∧ z 2 } {z 1 , z 2 }. If all the elements of D are below z 1 = C 0 ∨ C 1 , then k 0 ≤ C 0 ∨ D ≤ C 0 ∨ C 1 ,
contradicting the minimality of k 0 m C 0 C 1 {k 1 }. Therefore, at least one element of D is below z 2 = k 0 . If C 0 = ∅, then in the minimal join-cover C 0 D there are at least two elements that are below k 0 . This however contradicts (π-VarRL1), whence (VarRL1). We have, therefore, C 0 = ∅.

Lemma 8. If L is a pluperfect lattice with L |= [[AxRel]], then (π-StrongSymPC) holds in is OD-graph.
Proof. By Lemmas 5 and 7, (π-RMod) and (π-VarRL1) hold in the OD-graph. Let x := k 0 , y := C 0 , z := C 1 . Then the left-hand side of (SymPC) evaluates to k 0 which is therefore below the right-hand side. Thus, by Lemma 4, there is a minimal join-cover of the form k 0

m D 0 D 1 with either (i) D 0 m C 0 , D 0 {y ∧ (x ∨ z)}, and D 1 = C 1 , or (ii) D 0 = C 0 , D 1 m C 1 , and D 1 {z ∧ (x ∨ y)}.
W.l.o.g. we can suppose that (i) holds. From D 0 {y ∧ (x ∨ z)} {x ∨ z} = {k 0 ∨ C 1 }, we can argue as follows. We notice first that if an element of D 0 is join-prime, then it is either below k 0 or below some c ∈ C 1 ; since D 0 C 1 is an antichain, this element is below k 0 . Therefore, if all the elements of D 0 are join-prime, then, by (π-VarRL1), D 0 = {k }. Otherwise, there exists a non-joinprime element k in D 0 and, by (Unjp), this is the only non-join-prime in D 0 . Write D 0 = {k } E, then every element of E is join-prime and, as seen before, we need to have E {k 0 }. Then (π-RMod) enforces E 0 = ∅ and D 0 = {k }. In both cases, the relation {k }

= D 0 m C 0 yields k m C 0 .
Finally, in order to understand the structure of finite lattices in the variety of axiomatized by [[AxRel]], let J p (L) denote the set of join-prime elements of L and consider the following property:

• If k 0 m C and C ⊆ J p (L), then c 0 ≤ k 0 for some c 0 ∈ C (π-JP)
The next Lemma ensures the existence of a non-join-prime element in a cover in finite atomistic lattices, as stated in Theorem 7.

Lemma 9. If a finite lattice L satisfies [[AxRel]], then (π-JP) holds in its ODgraph. In particular, if L is atomistic, then k 0 m C implies that k 1 ∈ C for some non-join-prime k 1 .

It can be shown that the finiteness assumption in Lemma 9 is necessary.

Conclusions and further directions

Some undecidable problems. Our main result, Theorem 7, characterizes the OD-graphs of finite atomistic lattices satisfying [[AxRel]] as structures similar to frames for the commutator logic [S5] n , the multimodal logic with n distinct pairwise commuting S5 modal operators, see [START_REF] Kurucz | Combining modal logics[END_REF]. We exemplify next how to take advantage of such similarity and of the existing theory on combination of modal logics, to deduce undecidability results. As this is not the main goal of the paper, we delay a full exposition of these ideas to an upcoming set of notes. An [S5] n frame is a structure F = (F, R 1 , . . . , R n ) where each R i is an equivalence relation on F and, moreover, the confluence property holds: if i = j, xR i y and xR j z, then yR j w and zR i w for some w ∈ F . A particular class of [S5] n frames are the universal S5 n -products, those of the form U = (F, R 1 , . . . , R n ) with F = X 1 × . . . × X n and (x 1 , . . . , x n )R i (y 1 , . . . , y n ) if and only if x j = y j for each j = i.

For a frame F = (W, R 1 , . . . , R n ) and X ⊆ {1, . . . , n}, let us say that Y ⊆ W is X-closed if w 0 ∈ Y , whenever there is a path w 0 R i0 w 1 . . . w k-1 R i k w k with {i 0 , . . . , i k } ⊆ X and w k ∈ Y . Then X-closed subsets are closed under intersections, so subsets of {1, . . . , n} give rise to closure operators X and to an action as defined in Section 3. Let L(F) = P ({1, . . . , n}) j P (W ) and notice that L(F) is atomistic. A frame F is initial if there is f 0 ∈ F such every other f ∈ F is reachable from f 0 ; it is full if, for each i = 1, . . . , n, R i is not included in the identity. If F is initial and full, then L(F) is subdirectly irreducible. A p-morphism is defined as usual in modal logic. The key observation leading to undecidability is the following statement. Theorem 9. There is a surjective p-morphism from a universal S5 n -product frame U to a full initial frame F if and only if L(F) embeds in a relational lattice.

Proof (Sketch). The construction L is extended to a contravariant functor, so if ψ : U --→ F is a p-morpshim, then we have an embedding L(ψ) of L(F) into L(U). We can assume that all the components X 1 , . . . , X n of U are equal, so X i = X for each i = 1, . . . , n; if this is the case, then L(U) is isomorphic to the relational lattice R({1, . . . , n}, X).

The converse direction is subtler. Let χ : L(F) --→ R(A, D) be a lattice embedding; since L(F) is subdirectly-irreducible, we can suppose that χ preserves bounds; its left adjoint µ : R(A, D) --→ L(F) is then surjective. Since both L(F) and R(D, A) are generated (under possibly infinite joins) by their atoms, each atom x ∈ L(F) has a preimage y ∈ R(D, A) which is an atom. Consider now S 0 = {f ∈ D A | µ(f ) is a non-join-prime atom} and make it into a P ({1, . . . , n})valued ultrametric space by letting δ S0 (f, g) = µ(δ(f, g)) ⊆ {1, . . . , n}-we use here the fact that µ sends join-prime elements to join-prime elements. S 0 is shown to be a pairwise complete ultrametric space over {1, . . . , n}. We prove that pairwise complete ultrametric spaces over a finite set B are in bijection with universal S5 n -product frames, with n = card B. Then the restriction of µ to S 0 is a surjective p-morphism from S 0 to (a frame isomorphic to) F.

In view of the following statement, which relies on [START_REF] Hirsch | Representability is not decidable for finite relation algebras[END_REF] and can be inferred from [START_REF] Hirsch | On modal logics between K × K × K and S5 × S5 × S5[END_REF]: "for n ≥ 3, it is undecidable whether, given a finite full initial frame F, there is a surjective p-morphism from a universal S5 n -product U to F" , we deduce the following undecidability results, which partially answer Problem 4.10 in [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF].

Corollary 3. It is undecidable whether a finite subdirectly irreducible atomistic lattice embeds into a relational lattice. Consequently, the quasiequational theory of relational lattices in the pure lattice signature is also undecidable.

Comparison with Litak et al. [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF]. We have presented our first contribution to the study of the equational theory of relational lattices. In [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF] two equations in the larger signature with the header constant are presented as a base for the equational theory of relational lattices. As mentioned there, the four equations of [[AxRel]] are derivable from these two equations. Therefore, we can also think of the present work as a contribution towards assessing or disproving completeness of these two axiomatizations. Yet, we wish to mention here and emphasize some of our original motivations. Lattice theoretic equations are quite difficult to grasp, in particular if considered on the purely syntactic side, as done for example in [START_REF] Litak | Relational lattices: From databases to universal algebra[END_REF]. Duality theory attaches a meaning to equations via the combinatorial properties of the dual spaces. This process is nowadays customary in modal and intuitionistic logic and gives rise to a well defined area of research, correspondence theory. Our aim was to attach meaning to the equations of relational lattices. The answer we provide is, at the present state of research, via the relevant combinatorial properties, symmetry and pairwise completeness. From this perspective, the results presented in Section 6 undoubtedly need further understanding. In particular it is worth trying to modularize them, so as to discover equations exactly corresponding to symmetry or, respectively, to pairwise completeness; alternatively, argue that these equations do not exist. Finally, the present work opens new directions and challenges for the duality theory developed in [START_REF] Santocanale | A duality for finite lattices[END_REF]-of which, we hope we have illustrated the fruitfulness-including a better understanding of how to generalize it to the infinite case, new mechanisms by which to devise correspondence results, natural conjectures concerning equations having correspondents in finite lattices.

Definition 2 .

 2 Let L be a pluperfect lattice and let X, Y ⊆ J (L) be antichains. Put X m Y if X { Y } and y ∈ C xy for some x y ∈ X, for each y ∈ Y and whenever {C x | x ∈ X} is a family of coverings of the form x m C x Y .

Lemma 5 .

 5 If (VarRL1) holds on a pluperfect lattice, then its OD-graph satisfies (π-VarRL1).Proof. Suppose C = {k 1 } {k 2 } D with k 0 m C and k 1 , k 2 ≤ k 0 . Let x := k 0 , y := k 1 ∨ D, z := k 2 ∨ D. Then k 1 ≤ x ∧ y, k 2 ≤ x ∧ z and D ≤ y ∧ z,

Lemma 7 .

 7 If L is a pluperfect lattice such that L |= [[AxRel]], then (π-RMod) holds in its OD-graph. Proof. Let k 0 C m k 1 and put C = C 0 C 1 with C 0 {k 0 } and c ≤ k 0 for each element c ∈ C 1 . As (Sym) whence (π-Sym) hold, we have

This is a lifting of the Hamming distance to subsets. Yet, in view of[START_REF] Priess-Crampe | Equivalence relations and spherically complete ultrametric spaces[END_REF] and of their work on generalized ultrametric spaces, such a distance might reasonably tributed to Priess-Crampe and Ribenboim.