Marco Fedi
email: marco.fedi@istruzione.it

Mario R L Artigiani
email: artigiani@engineering3d.it

Theoretical and CFD analysis of gravity and galaxy formation

Keywords: dilatant vacuum, quantum gravity, spin, galaxy formation, flat profile, Higgs

come

Introduction

The vacuum as a superfluid has been studied in several approaches [START_REF] Volovik | The Superfluid Universe[END_REF][START_REF] Sbitnev | Physical Vacuum is a Special Superfluid Medium[END_REF][START_REF] Huang | A Superfluid Universe[END_REF], whose recent evolution is for instance the logarithmic BEC vacuum theory [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF][START_REF] Avdeenkov | Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent[END_REF]. Nevertheless, the hydrodynamic characteristics of the vacuum seem to be much different in various cases. The Nobel laureate R.B. Laughlin, states:

Studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness. It is filled with stuff that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. This behavior is typical of non-Newtonian shear-thickening fluids (dilatant fluids): they are fluid when low shear stress is applied and they progressively solidify as shear stress increases. The situation described by Laughlin, as regards large particle accelerators, has been in fact described as due to the existence of a dilatant vacuum [START_REF] Fedi | Relativistic mass due to a dilatant vacuum leads to a quantum reformulation of the relativistic kinetic energy[END_REF], through which charges accelerate. Not by case, by treating the vacuum as a shear-thickening fluid, the relativistic perihelion precession of the planet Mercury has been solved by deriving Einstein formula for the precession of perihelia in a simpler way and the Pioneer anomaly has been directly computed, with much greater accuracy, by using the Lorentz factor in Stokes' law, as a viscosity factor of the vacuum [START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF]. In this way the curve expressed by the Lorentz factor turned out to be the rheogram of the vacuum. Vacuum dilatancy arises both with particles and with macroscopic bodies (probes, planets) traveling through a vacuum. In this study, the analysis of the vacuum as a dilatant fluid is continued as regards gravity and galaxy formation and rotation. We start from the vacuum in superfluid regime, by describing fundamental fermions as toroidal vortices (Sect. I). In the second section we derive an equation for quantum gravity, without the classical Newtonian constant. The quantum aspects of this formula are then discussed in Sect. III. In the following section we simulate a quadrupole via computational fluid dynamics (CFD) and we show that gravitational waves can be described as negative pressure waves in a dilatant vacuum (Sect. IV). Further CFD simulations demonstrate that a binary black hole located in the center of a galaxy and in a dilatant vacuum, can generate various galaxy shapes (Sect. V) a well as the flat profile of the rotation velocity (Sect. VI). Eventually, we describe an unexpected effect obtained in the simulations: a breath of galaxies (Sect. VII).

I. Spin-generated Bernoulli force as the engine of gravity.

Dilatant vacuum can be interpreted as a superfluid (dark energy [START_REF] Amendola | Dark Energy. Theory and Observations[END_REF]) doped with scattered dark-matter particles [START_REF] Bertone | Particle Dark Matter. Observations, models and searches[END_REF], which together make up ∼ 95% of the mass-energy of the universe: something like a cosmic oobleck [START_REF] Kliever | Describing space as oobleck solves decades-old physics mystery[END_REF]. The dilatant-vacuum model therefore goes in the direction of a unification of dark energy and dark matter into a single dark fluid [START_REF] Arbey | Is it possible to consider Dark Energy and Dark Matter as a same and unique Dark Fluid?[END_REF][START_REF] Farnes | A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified Λ CDM Framework[END_REF]. At the same time, vacuum dilatancy could be also interpreted as a hydrodynamic feature of the viscous Higgs field. It has been proved that the apparent viscosity of the vacuum obeys the Lorentz factor [START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF], thus, when shear stress is low, the vacuum is in a superfluid-like regime. Just like classical mechanics is placed at the beginning of the curve expressed by the Lorentz factor. The temperature of the superfluid vacuum at 2.72 K, i.e. that of the cosmic microwave background (CMB), would match that of laboratory superfluids. In superfluids, quantized vortices spontaneously arise and since a vortex tube must extend to a boundary or close in a loop (an interesting connection with loop quantum gravity [START_REF] Rovelli | Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory[END_REF] can be seen in this fact), obeying Helmholtz's second principle, vortex rings (vortex tori) can form in a superfluid vacuum (Fig 1) [START_REF] Sbitnev | Physical Vacuum is a Special Superfluid Medium[END_REF][START_REF] Huang | A Superfluid Universe[END_REF][START_REF] Jehle | Topological characterization of leptons, quarks and hadrons[END_REF]. However, what is also important is the fact that only vortices in a doped superfluid show correct attraction-repulsion interactions among them [START_REF] Pshenichnyuk | Pair interactions of heavy vortices in quantum fluids[END_REF] (mathematically corresponding to gravity or electromagnetic interactions). Thus, to describe a fundamental fermion as a toroidal vortex in a superfluid vacuum, it is necessary to specify that this superfluid has to be doped. The dilatant-vacuum picture is therefore fit for the purpose, since a dilatant is made up of a (super)fluid and scattered dopant particles. For example scattered dark matter in a sea of superfluid dark energy.

Let us analyze the fundamental equations of what above described.

Considering the physical vacuum as a doped Bose-Einstein condensate (BEC), we start from the Gross-Pitaevskii equation (GPE) to analyze its hydrodynamic behavior [START_REF] Dalfovo | [END_REF] ih

∂ ψ ∂t = - h 2 2m ∇ 2 ψ + gψ |ψ| 2 -ψ µ 0 ( 1
)
where ψ is the condensate wave function, m the mass of a quantum of the fluid vacuum, µ 0 the chemical potential and g = 4πah 2 /m a low-energy parameter, where a represents the scattering length between vacuum's quanta. In the phase representation ψ = √ ρ / 0 e iϕ , ρ / 0 = |ψ| 2 is vacuum's density. From (1) we can write the continuity equation and the analogue of the Euler equation

∂ ρ / 0 ∂t + ∇ • (ρ / 0 v S ) = 0 (2) m ∂ ρ / 0 ∂t + v S • ∇ v S = ∇ µ 0 -gρ + h2 2m ∇ 2 √ ρ / 0 √ ρ / 0 (3)
where v S = h m ∇ϕ is the superfluid velocity and h2 2m

∇ 2 √ ρ / 0 √ ρ / 0 (4)
the quantum potential. The condensate must be a continuous function in space, so its phase is continuous modulo 2π. We define the quantized circulation (Γ) with the line integral

C dx • v S = 2π h m n ≡ Γ (n = 0, ±1, ±2, ...) (5)
where C is a close loop in space, encircling a vortex line, ψ = 0, and the superfluid's density vanishes. The problem of ultraviolet divergence and of the radius of the fundamental particles is solved in this approach by considering toroidal vortices of vacuum's quanta instead of point-like particles, whose radius is r = 2ξ , with ξ defined as the healing length,

ξ ≡ V 8πaN , ( 6
)
where V is the volume in which the vortex arises, a the scattering length and N a normalized number of vacuum's quanta in the volume. A hydrodynamic reformulation of the Barut-Zanghi theory [START_REF] Salesi | Hydrodynamical reformulation and quantum limit of the Barut-Zanghi Theory[END_REF], which includes spin, was proposed by Salesi and Recami, who suggested [18], similarly to us, that quantum potential

Q = - 1 2 m v 2 S - 1 2 ∇ • v S (7)
(in natural units, h = 1) totally arises from the internal motion of a particle v S × s, where

v S = 1 2m ρ -1 ∇ρ = 1 2m ∇R 2 R 2 , (8)
being s the direction of spin. The link to the concept of vortex-particle is, in this way, quite immediate. If, to complete a turn in the poloidal direction, a quantum in the vortex needs the same time the vortex needs to complete two turns in the toroidal direction, then the vortex has spin ½ (fermion), i.e. the system returns to the same state after a rotation of 720°a nd after each quantum in the vortex has traveled along a Möbius-strip path (Fig. 1). By defining ω 1 , ω 2 as the angular velocities (toroidal and poloidal), the spin angular momentum (S) is determined by the ratio ω 1 /ω 2 = (nπ/dt) / (2π/dt) and after the cancellations

ω 1 ω 2 = n 2 = S. (9)
Spin-0 vortices may arise through a further evolution of the torus into a spheroidal vortex. The ratio of the angular velocities ω 1 and ω 2 of vacuum's quanta in the vortex (Fig. 1) satisfactorily represents all spin numbers. The parametric equation defining the position of a quantum in the torus vortex is

     x = (r + ξ cos (ω 1 dt + φ 0 )) cos (ω 2 dt + θ 0 ) y = (r + ξ cos (ω 1 dt + φ 0 )) sin (ω 2 dt + θ 0 ) z = ξ sin (ω 1 dt + φ 0 ) (10)
where r ≥ 2ξ is the distance between the centers of the tube and of the torus, ω 1 = dφ /dt, ω 2 = dθ /dt and φ 0 and θ 0 are phases with arbitrary values between 0 and 2π. The toroidal shape of fundamental particles also resembles the loops of Loop Quantum Gravity [START_REF] Rovelli | Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory[END_REF][START_REF] Ashtekar | Loop Quantum Gravity[END_REF] but in our case on a higher scale to describe a fundamental fermion. At the same time, we can also notice a certain similarity between vortex lines and string theory. Vacuum's hydrodynamics has in our opinion the potential to reconcile different approaches into a single model. Vortex chirality represents matter-antimatter parity (vortex-antivortex annihilation occurs and phonons are generated) and an opportune change in the ratio ω 1 /ω 2 would transform a boson into a fermion, or vice versa: such a mechanism could be therefore important for better understanding the foundations of supersymmetry.

It is known that quantum vortices exert Bernoulli force, generating attractive or repulsive forces [START_REF] Huang | A Superfluid Universe[END_REF][START_REF] Pshenichnyuk | Pair interactions of heavy vortices in quantum fluids[END_REF]. In particular, Pshenichnyuk proved that vortices in a doped superfluid behave differently, compared to those in a standard superfluid [START_REF] Pshenichnyuk | Pair interactions of heavy vortices in quantum fluids[END_REF]. The correct attractive or repulsive force, obeying an inverse-square law, is observed only in doped superfluids. The kinetic energy density, acting on the dopant's particles and caused by the superposition of the velocity fields of the superfluid vortices, can be written as

K ++ = h2 2m Q ψ 2 ∞ 4R 2 + d 2 -4Rd cos α (R 2 + ξ 2 ) (R 2 + d 2 + ξ 2 -2Rd cos α) , (11)
K +-= h2 2m Q ψ 2 ∞ d 2 (R 2 + ξ 2 ) (R 2 + d 2 + ξ 2 -2Rd cos α) , (12)
where the + + / +signs refer to same or different topological charges denoting repulsion or attraction, d is the distance between the vortices, R the radius of a doping particle, ξ the healing length and α the azimuthal angle in cylindrical coordinates which is associated with the doping particle. This mechanism is driven by the Bernoulli force, whose equation is

F b = S K(r)n(r)dS (13)
where K(r) = ρv 2 /2 expresses the kinetic energy density, which dominates on the vortex surface, while the density of the superfluid drops to zero within the healing length, and n(r) is a unit vector, normal to the surface S, over which the integral is calculated. Bernoulli force arises in superfluids as a superposition of the velocity fields of the vortices.

Points where the fields reinforce each other and where their interaction makes them weaker produce a pressure-related Bernoulli effect, that is, in our opinion, the mechanism of quantum gravity. Eq. ( 13) mathematically equals Gauss's law for gravity

F g = S g(r)n(r)dS. (14)
For our approach, this equation is much more convenient than Newton's law (albeit it is equivalent to it), because it describes the gravitational field as a flux entering into a spherical volume. We start by hypothesizing that this flux is due to a real flow [START_REF] Kirkwood | The Physical Basis of Gravitation[END_REF][START_REF] Cahill | Gravity as quantum foam in-flow[END_REF] of fluid, dilatant vacuum, caused by the spin-mediated bernoulli effect. In short, the equation for gravity [START_REF] Jehle | Topological characterization of leptons, quarks and hadrons[END_REF] is the expression of the Bernoulli force [START_REF] Rovelli | Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory[END_REF] and this can lead, as discussed below, to the quantization of gravity. Of course, also macroscopic bodies, such as a planet, which consist of vortex-particles (fundamental fermions assembled in nuclei, atoms and molecules), produce a pressure gradient around themselves, due to the Bernoulli effect, which is called gravitational field (Fig. 2). It is interesting to consider that toroidal vortices, whose rotations conflict, annihilate if they come into contact: parity symmetry and matter-antimatter annihilation would be then easily explained in hydrodynamic terms and since vortex-antivortex annihilation produces, as we know, phonon emission, the photons emitted in matter-antimatter annihilation are in this picture phonons propagating through the dilatant vacuum's quasi-lattice: they propagate transversally and at a very high frequency (two peculiar characteristics of light), thanks to the shear-thickening feature of the vacuum.

II. Equation of quantum gravity without classical gravitational constant.

The macroscopic Bernoulli effect, detectable as the gravitational field of large bodies, is thence a pressure gradient which causes acceleration toward the center of the massive object, according to the known hydrodynamic law

a = -∇ P ρ ( 15
)
where a is the acceleration, P the pressure and ρ the density of the fluid. By applying [START_REF] Pshenichnyuk | Pair interactions of heavy vortices in quantum fluids[END_REF] to dilatant vacuum (denoted by the subscript / 0) we can write the expression for the gravitational field

g = -∇ P / 0 ρ / 0 ( 16
)
It is fundamental to notice that, in this new expression of the gravitational field, the Newtonian constant of gravitation disappears. Here we compare the formulas (with units) of the classical and quantum (Sect. III) gravitational potential

V = -G M r m 2 s 2 ⇔ V Q = - P / 0 ρ / 0 m 2 s 2 . ( 17
)
Generally in physics the presence of a constant refers to the necessity of adjusting the scale and the measurement units. In this case, Newton's formula for universal gravitation resorts to quantities such as mass and distance that are not directly involved in quantum gravity, a force that actually requires to consider pressure and density of the vacuum [START_REF] Salesi | Hydrodynamical reformulation and quantum limit of the Barut-Zanghi Theory[END_REF]. The fact that such a constant is used by Einstein in general relativity (GR) underlines a limitation of the theory, which is not indeed a quantum theory of gravitation. After a century from GR and considering the correct alternative solution offered by the dilatant-vacuum approach to the first classical test of GR, i.e. to the perihelion precession of the planet Mercury [START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF], we believe the time has come to switch to pressure, density and viscosity in a fluid dilatant vacuum, as a hydrodynamic process occurring over time, instead of the mathematical concept of curvature in a space-time, that, differently from the dilatant-vacuum approach, cannot for instance justify the flat profile of galaxy rotation velocity (Sect. VI): the various positive results achieved thus far as regards the existence of a dilatant space cannot be ignored. By replacing the new expression for the gravitational field g (16) in the formula of gravitation F g = -m g, we directly obtain an equation for quantum gravity

F Q g = -m ∇ P / 0 ρ / 0 , (18)
(the superscript Q means quantum) whose quantum nature is analyzed in the following section.

III. Quantum aspects

The classical gravitational potential energy U = -GMm/r becomes then

U Q = -m P / 0 ρ / 0 = F Q g ∇ . ( 19
)
Let us demonstrate that U Q is the quantum potential of (18). We first express ( 19) by resorting to momentum

U Q = -m P / 0 ρ / 0 = -p • u = - p 2 m , (20)
where u = p /m. From ( 20), switching to the momentum operator, we can write a time-independent, one-dimensional, eigenvalue Schrödinger equation (SE)

p2 m ψ = - h2 m ∇ 2 ψ = - h2 m 1 r 2 ∂ ∂ r r 2 ∂ ∂ r ψ = = V (r)ψ ( T ψ=0) = Ĥψ = Eψ, (21)
considering no initial velocity. The expectation value for the gravitational potential energy is

E = p 2 m = ∞ 0 ψ * - h2 m 1 r 2 ∂ ∂ r r 2 ∂ ∂ r ψ dr. ( 22
)
The velocity field associated with the pressure field generated by the vortex-particles (grouped for instance in a macroscopic massive object), via Bernoulli force [START_REF] Rovelli | Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory[END_REF], in the fluid dilatant vacuum acts as a pilot wave in the vacuum, driving any particle through the gravitational field. It may be useful to resort to Madelung's quantum hydrodynamic approach. Let us perform polar decomposition of the wavefunction,

ψ = √ ρ / 0 e i h S
, being S the phase. By replacing into the SE and introducing time dependency, we obtain the Madelung equations, that is the continuity equation

∂ t ρ / 0 + 1 m ∇ (ρ / 0 ∇S) = 0, (23)
that describes the flow of vacuum's quanta along the current in the fluid vacuum, produced by the Bernoulli effect, and the Hamilton-Jacobi equation

∂ t S = - 1 2m (∇S) 2 -V (r) -Q, (24)
where

Q = - h2 2m ∇ 2 √ ρ / 0 √ ρ / 0 ( 25
)
is the quantum potential. Since we are considering no initial velocity and no classical potential (being the gravitational potential explained through the described quantum picture), the action S over time, i.e. the acceleration of a particle in the velocity field, reduces to

∂ t S = -Q. ( 26
)
Thus the quantum-hydrodynamic interpretation of the classical gravitational potential corresponds to double the quantum potential expressed in units of energy

U Q = -m P / 0 ρ / 0 = 2Q = - h2 m ∇ 2 √ ρ / 0 √ ρ / 0 , (27)
and ( 18) can be simplified to

F Q g = 2∇Q, ( 28
)
as double the corresponding quantum force (∇Q), which can be expressed also in function of the quantum pressure tensor p Q , where pressure in a dilatant quantum vacuum replaces the curvature of Einstein's geometrical space-time, specifically the curvature expressed by the Einstein tensor G µυ = R µυ -2 -1 g µυ R, of the field equations, where R µυ is the Ricci tensor and R the scalar curvature. Resorting to the quantum pressure tensor, quantum gravity can be expressed as a hydrodynamic quantum force in a dilatant quantum vacuum

F Q g = - 2m ρ m ∇ • - h 2m 2 ρ m ∇ ⊗ ∇ ln ρ m , (29)
where ρ m = mρ is mass density.

IV. Gravitational waves as negative pressure waves in a dilatant vacuum

Being in our model a non-Newtonian dilatant fluid, the vacuum can be treated as a quasi-classical quantum fluid. Quantum-like gravity waves in a classical fluid have been described by Nottale [START_REF] Nottale | Quantum-like gravity waves and vortices in a classical fluid[END_REF]. By resorting to CFD we simulated the action of a quadrupole in the fluid dilatant vacuum (3). The result is the propagation of negative pressure waves corresponding to gravitational waves, once the currently adopted geometrical explanation of GR is replaced by our hydrodynamic model. Indeed, if a distant test mass were hit by these negative-pressure waves propagating in the vacuum, it would be expected to exactly reacts as if it were hit by gravitational waves. This because we justify the gravitational force as the Bernoulli force in a fluid vacuum (Sections I). The simulation therefore shows gravitational waves as actually the propagation of negative pressure waves in the fluid vacuum (Fig. 4). Let us consider a supposed space-time deformation

h × = - 1 R G 2 c 4 4m 1 m 2 r (cos θ ) sin 2ω t - R c . ( 30
)
By replacing G from [START_REF] Salesi | Hydrodynamical reformulation and quantum limit of the Barut-Zanghi Theory[END_REF], that is G = (rP / 0 /Mρ / 0 ), where M = m 1 + m 2 , and resorting to (ρ / 0 j / 0 ) -1/2 = (ε 0 µ 0 ) -1/2 , where j / 0 is the shear compliance of the shear-thickening vacuum and ε 0 and µ 0 are the permittivity and permeability (respectively) of the vacuum, we see that a gravitational wave is a pressure oscillation propagating through the dilatant vacuum at the speed of light (phonon speed in a dilatant vacuum))

h × = - r R (2P / 0 j / 0 ) 2 (m 1 + m 2 ) -1 (cos θ )• • sin 2 V Q r t -R ρ / 0 j / 0 , (31)
whereas the polarization h + reads where R is the distance from the observer, t the elapsed time, θ the angle between the perpendicular to the plane of the orbit and the line of sight of the observer, r the radius of the quadrupole, V Q /r = ω its angular frequency, obtained by resorting to the identity in [START_REF] Salesi | Hydrodynamical reformulation and quantum limit of the Barut-Zanghi Theory[END_REF] in the Newtonian formula for constant angular velocity of a circular orbit G(m 1 + m 2 )/r 3 . A variation in pressure causes acceleration, acting for example on LIGO's test masses.

h + = -2 r R (P / 0 j / 0 ) 2 (m 1 + m 2 ) -1 (1 + cos 2 θ )• • cos 2 V Q r t -R ρ / 0 j / 0 , (32)

V. Galaxy formation and shapes

We have applied the above-described simulation also to galaxy formation, by considering a binary black hole in the galactic core. By varying pressure and angular velocity we obtained different shapes for galaxies. This suggests that also galaxy formation may be driven by pressure gradients in a dilatant vacuum, due to a rotating, anisotropic mass distribution, as in a binary system. We can therefore infer that two or more black holes in the core can generate different kinds of spyral galaxies [START_REF] Hailey | [END_REF], whereas a single supermassive black hole could be responsible for the shape of elliptical galaxies.

It is important to note that galaxies generated in the dilatant In (e) and (f) pressure is -100 Pa and sectors of 45°are used (differently from the 90°sectors showed in Fig. 3). In (f), with respect to (e), representing a grand-design galaxy, angular velocity has been halved, so an anemic galaxy appears.

vacuum show to possess a flat profile of the rotation velocity, according to observations (simulation of galaxy evolution at [START_REF] Fedi | film of CFD simulation describing the birth, the evolution and the behavior of a spyral galaxy in a fluid, dilatant vacuum[END_REF]). This means that neither current dark matter theory [START_REF] Trimble | Existence and nature of dark matter in the universe[END_REF][START_REF] Kuijken | The Mass Distribution in the Galactic Disc -Part III -the Local Volume Mass Density[END_REF] nor MOND are necessary to explain the flat profile but it is enough to replace geometrical Einsteinian space-time with a fluid, dilatant vacuum, in which hydrodynamic phenomena driven by pressure and viscosity occur over time, which is relative also in our approach, due to clocks ticking in a more or less viscous vacuum (being apparent viscosity related to shear stress). A clock traveling faster through the dilatant vacuum undergoes greater apparent viscosity according to vacuum's rheogram, i.e. to the Lorentz factor. In short, GR is generally a correct quantitative tool (not for galaxy rotation for instance, whereas the dilatant vacuum works properly, see Sect. VI) but its qualitative explanation (space-time deformations) actually corresponds to the action of pressure gradients and apparent viscosity in a dilatant vacuum, as demonstrated in [START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF]. By analyzing the different galaxy types showed in Fig. 5, we FIG. 6. Pressure in the dilatant vacuum along the X-direction of the simulations. The letters of the plots correspond to those of the galaxy shapes in Fig. 5. Animated plots at [START_REF] Fedi | film of CFD simulation describing the birth, the evolution and the behavior of a spyral galaxy in a fluid, dilatant vacuum[END_REF] notice that a weaker pressure gradient, due to less baryon matter in the galaxy core, generates a smaller galaxy: Fig 5 c) represents a dwarf spyral (dS) and letter d) a pea galaxy. In e) we see a grand-design galaxy, whereas in f) an anemic galaxy, obtained by halving the angular velocity of the galaxy. In a spyral galaxy, stars tend to concentrate along the low-pressure corridors (the galaxy arms) in the dilatant vacuum, as the fluid flows toward the black holes situated in the galaxy core. This result from CFD is compatible with the findings presented by Martin, O'Sullivan et al. [START_REF] Martin | Multi-filament gas inflows fuelling young star-forming galaxies[END_REF], who described multifilament gas inflows fuelling galaxies: indeed, not only stars but of course also gases can flow along such pressure corridors in the fluid vacuum. The specific black-holes configuration in a core is another element which determines the shape of the galaxy. A 45°configuration (Fig. 5 letter e.) increases the grand-design feature of the galaxy and its size. But when the angular velocity is halved, the grand-design converts into a so-called anemic galaxy. Correct stars migration over time from an arm to another of a spyral galaxy has been also observed in the CFD simulations (Fig. 7). A CFD simulation showing the birth, the evolution and the behavior (rotation, FIG. 8. To confirm the flat profile of the rotation curve we monitored the angular velocity of some probes at different arbitrary radii (R1 to R13). The positive result highlighting the flat profile is shown in Fig. 9 breath) of a spyral galaxy in a dilatant vacuum has been realized by us and it is available at [START_REF] Fedi | film of CFD simulation describing the birth, the evolution and the behavior of a spyral galaxy in a fluid, dilatant vacuum[END_REF].

VI. Flat profile obtained in a dilatant vacuum

The galaxies obtained in our simulations show a flat profile of angular velocity, according to astronomical observations. This implies that the vacuum as a dilatant fluid (a unified dark fluid that, due to its viscosity and ubiquity may correspond to the Higgs field itself, further extending the unification) is a sufficient condition to justify the flat profile. We simulated the motion of probes along concentric orbits in the galaxy, at six different radius values (Fig. 8), to verify the dependence of the rotation velocity on the distance from the galactic core. See also [START_REF] Martin | Multi-filament gas inflows fuelling young star-forming galaxies[END_REF] We confirm (Fig. 9) that the rotation frequency does not vary with distance, implying a flat profile of the rotation velocity. Another important consideration concerning the existence of a dilatant vacuum, is that it can justify the expansion of the universe without bringing into play the concept of negative mass, because the energy density of the vacuum corresponds to pressure (J/m 3 = Pa), causing expansion. For these reasons, this study and the previous ones [START_REF] Fedi | Relativistic mass due to a dilatant vacuum leads to a quantum reformulation of the relativistic kinetic energy[END_REF][START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF], that investigate the specific nature of physical vacuum as a dilatant fluid, are in our opinion of crucial importance.

VII. Breath of galaxies

In the video simulation in [START_REF] Fedi | film of CFD simulation describing the birth, the evolution and the behavior of a spyral galaxy in a fluid, dilatant vacuum[END_REF], one can watch the birth, evolution and behavior of a spyral galaxy in the fluid, dilatant vacuum. The animated diagram on the right-hand side of the video, shows the persistence of the flat profile of the rotation velocity over time, during the evolution and the life of the galaxy. It is also interesting to see that a breath of the galaxy emerges, as pressure waves traveling from the galaxy core cross the galactic disk that then continue to propagate through the intergalactic dilatant space. We enlarged the simulation domain enough to check whether the effect could vary and depend on the boundary conditions, but it persisted.

We believe this result could be linked to the phenomenon described in [START_REF] Widrow | Galactoseismology: discovery of vertical waves in the galactic disk[END_REF].

VIII. Conclusion

The theoretical and computational conclusions of this study reinforce the dilatant-vacuum model, suggesting that vacuum dilatancy could be a hydrodynamic property of the Higgs field itself, which is indeed endowed with a certain viscosity: if this is the case, we are thence showing something unexpected, i.e. that the Higgs field also interacts with macroscopic bodies (for instance with the planet Mercury and the Pioneer probes [START_REF] Fedi | Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury's perihelion precession[END_REF]) and that this ubiquitous fluid is enough to justify the formation and the specific rotation of galaxies, besides being a key to understanding quantum gravity, by describing the gravitational field as an inflow of dilatant vacuum (of Higgs fluid?) into massive particles, driven by the Bernoulli force produced by spin, as discussed in Sect. I. Alternatively, the dilatant vacuum can be seen as a unified, viscous dark fluid, i.e. superfluid dark energy and scattered dark matter particles that act as a dopant and cause space dilatancy as shear stress increases. What we actually believe is that a dilatant vacuum could allow an extended unification of Higgs field, dark energy and dark matter. Our simulations also showed that a binary black hole can be the cause both of gravitational waves as negative pressure waves through the dilatant vacuum and of the formation of various types of spyral galaxies. A noteworthy effect similar to a breath of galaxies emerged from the simulations, whose cause is not yet clear but could reconnect to previous studies concerning galactoseismology by Widrow, Gardner and collegues [START_REF] Widrow | Galactoseismology: discovery of vertical waves in the galactic disk[END_REF]. Studies based on the equations of dilatant vacuum should be therefore continued, also as far as other phenomena and unanswered questions are concerned.

FIG. 1 .

 1 FIG. 1.A vortex line with healing length ξ , actually a vortex tube, closes in a circle, becoming a vortex torus. Below: the Möbius-strip trajectory of a hypothetical vacuum's quantum in the vortex torus: the quantum returns to any initial position after a rotation of 720°of the torus, so the ratio rotations/revolutions represents spin ½ (fermion). Below, on the right-hand side, we see the paths of 12 quanta.

FIG. 2 .

 2 FIG. 2. Above: gravitational field as pressure gradient in a fluid quantum vacuum, which obeys the inverse-square law. Below: velocity field due to the pressure gradient. Pressure and velocity scales are arbitrary.

FIG. 3 .

 3 FIG. 3. CFD simulation of a binary black hole. Above: rotating CFD subdomain with sectors of 90°a

FIG. 4 .

 4 FIG. 4. CFD simulation of gravitational waves as pressure waves in a fluid dilatant vacuum (arbitrary pressure scale). The black dot in the center is structured to mime a binary system in the dilatant vacuum, according to Fig 3

FIG. 5 .

 5 FIG. 5. CFD simulations of different kinds of galaxies: (a),(b),(c) and (d) represent galaxies at different arbitrary pressure gradients in the fluid vacuum: -100 Pa, -50 Pa, -25 Pa, -11 Pa (respectively).In (e) and (f) pressure is -100 Pa and sectors of 45°are used (differently from the 90°sectors showed in Fig.3). In (f), with respect to (e), representing a grand-design galaxy, angular velocity has been halved, so an anemic galaxy appears.

FIG. 7 .

 7 FIG. 7. Velocity arrows showing the direction in which stars migrate, along with galactic gases, from an arm to another of a spyral galaxy, due to the warped pressure field in the dilatant space.

FIG. 9 .

 9 FIG. 9. Pressure function of angle (for each radius). Since for the considered radii the wavelength is stable with good approximation, a flat profile of the rotation velocity is revealed.


Macarena Flores González

Alain Kuzniak
email: alain.kuzniak@univ-paris-diderot.fr

Elizabeth Montoya Delgadillo

Laurent Vivier
email: laurent.vivier@univ-paris-diderot.fr

Laurent Vivier Two

Two situations for working key properties of R

Keywords: Completeness, real numbers, mathematical analysis, algebra, transition to university

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Teaching and learning real numbers is not an easy topic, and it is probably for this reason that most secondary school curricula do not develop it, even if they are used. Several studies have been conducted on this subject, but often focusing on the double decimal representation of decimal numbers or on irrationality [START_REF] Oktaç | Conversion, change, transition…[END_REF] made a recension of researches on these theme). More recently, the emphasis has been placed on the completeness of R, for the topology of the order, as one of the key characteristic properties to be achieved (e.g. [START_REF] Bergé | Students' perceptions of the completeness property of the set of real numbers[END_REF][START_REF] Durand-Guerrier | Conceptualization of the Continuum, an Educational Challenge for Undergraduate Students[END_REF]. It has been hypothesized that these properties of R are important for understanding real analysis. We are in line with this hypothesis with a positioning in favour of an algebraic approach, but how to develop the simultaneous study of algebraic and topological properties of R? In particular, we have built situations that may introduce the completeness of R, on the basis of the algebraic properties of R. In the two preliminary studies we present here, we analysed the students' mathematical work using the Mathematical Working Space theory -MWS- [START_REF] Kuzniak | Mathematical working spaces in schooling: An introduction[END_REF].

Calculations with real numbers

This situation presents the basic algebraic operations in relation to the completeness of R. In fact, the definitions of the sum and the product in R are based on the completeness of R. They are rarely worked at the secondary level, as if these definitions were trivial.

We asked the students to find the roots x 1 , x 2 , x 3 of the polynomial P(x) = x 3 60x 2 +980x 4700 using adjacent sequences of decimal numbers. At this educational level, it is only possible to calculate approximate values of these roots since exact values need more advanced mathematics (unless using an algebra calculator). We, then, asked them to calculate the results of the basic operations on the three real roots, such as x 1 +x 2 +x 3 , that is actually 980 with the root properties of a polynomial. The visualization process is based on increasingly precise approximations (of the adjacent sequences) defining small nested intervals, and the signs (such as the tables of values), are used to support these processes. The main goal was to encourage work of visualization to conjecture the limits of each adjacent sequence, to recognize the polynomial coefficients.

This situation has been proposed to 6 grade 12 students by pairs, in France in 2018. The data are made on the student's dialogues and their written productions. Finally, the visualization of the limits did not take place, but this did not prevent elements of completeness from appearing spontaneously, as for the convergence and the intersection of the nested intervals, or from students confronting the epistemological obstacle of the potential infinite and the actual infinite.

Functional equation of the real exponential

The fact that it is hard to distinguish the density and completeness of R by graphical representation (especially in technological tools). In this sense, we propose to construct algebraically a function that allows a visualization of the discrepancy between density and completeness. This situation is attractive because it allows using and connecting of a wide range of academic knowledge.

The mathematical task consists in searching for the functions f defined on E, where E is the usual numbers sets N, Z, D and/or Q and R, with values in R, which satisfy the conditions: (a) f(1) = 2 and (b) ∀x ∀y on E f(x+y)=f(x)•f(y). The first question proved that f > 0 and that f(0)=1. Then, there is a unique function (the exponential) in N, Z, D and Q but this is not the case in R. More specifically, we ask for certain values and graphical representations. The functional equation is used as an artifact by calculating the values of f algebraically, and a graph is drawn for each set.

A first study in Chile with 6 students by pairs was conducted in 2017. The data are made of written students' answers and direct observation. For sets N and Z, there were some technical problems in using (b) to compute values and some students drew a continuous line but quickly realize that the context is discreet. For D and Q, the discussion is based on the line drawn and the density: do we represent a continuous line? Finally, a significant obstacle arose for R because of the need of topological arguments. We think that asking more explicitly to build a function f on Q.1+Q.3 (additive subgroup of R -1.a+3.b; a\in Q, b\in Q) with f(3)=1 could help, and so do the use of a technological software like Geogebra.

Conclusion and discussion

We presented two preliminary studies with these situations that have a great potential. Nevertheless, it seems that the introduction of an algebraic work allows to clearly show the topological jump made when "passing" from Q (or D) to R. Moreover, R cannot be reduced to its sole property of completeness: the uniqueness of R is based on several other properties, like its algebraic structure.

MWS theory has allowed us to design these situations and interpret the results obtained. Thus, the three dimensions of MWS has been activated: the work is done on the visualized signs (semiotic dimension), on the mathematical knowledge itself (discursive dimension) and on the technological tools to produce signs (instrumental dimension).


Valentin A B Küchle
email: kuechlev@msu.edu

Shiv S Karunakaran

Beliefs about learning attributed to recognized college mathematics instructors

Keywords: Beliefs, learning, teaching, college mathematics

The study of teachers' beliefs has a rich tradition [START_REF] Calderhead | Teachers: Beliefs and knowledge[END_REF], and several key categories of beliefs have been identified: teachers' beliefs about mathematics and the learning and teaching thereof [START_REF] Ernest | The impact of beliefs on the teaching of mathematics[END_REF]. Although the study of beliefs has largely focused on K-12 teachers, researchers have sought to extend this work to instructors of collegiate mathematics (e.g., [START_REF] Speer | Connecting beliefs and practices: A fine-grained analysis of a college mathematics teacher's collections of beliefs and their relationship to his instructional practices[END_REF]. Our study adds to this body of literature by focusing on a particular subset of college mathematics instructors, namely those who have been recognized with teaching awards. Although we do not equate winning a teaching award with teaching excellence, we note that recognized instructors are nominated by fellow colleagues and/or students, have had cause to reflect on their own teaching, and by winning the award have had their teaching explicitly commended by their department and college. Such an award adds weight to their voices in matters of teaching and makes them teaching role models. Thus, we seek to explore recognized instructors' beliefs about mathematics and the learning and teaching thereof.

Below, we present our study's preliminary findings from a series of interviews we conducted with recognized college mathematics instructors. We align with [START_REF] Ernest | The impact of beliefs on the teaching of mathematics[END_REF] conceptualization of beliefs and mirror his focus on beliefs about mathematics and the teaching and learning thereof. Furthermore, we espouse [START_REF] Speer | Issues of methods and theory in the study of mathematics teachers' professed and attributed beliefs[END_REF] view that researchers attribute beliefs to participants and concur that interviews alone do not do justice to our participants' beliefs. Thus, we see our interviews as forming the first step of a larger study including detailed classroom observations.

Method

At a large US institution, six mathematics instructors who had won at least one teaching award each were randomly chosen-from the set of thirteen mathematicians who had won a teaching award within the last five years-and individually interviewed for around one hour using a semistructured interview protocol. Two of the participants were non-PhD mathematicians, the other four held PhDs in mathematics. One of the non-PhD mathematicians was male (Aleph), the other female (Beth), whereas the four PhD mathematicians were all male (Gimel, Dalet, Waw, and Zayn).

All interviews were transcribed and coded with the help of MAXQDA, a qualitative data analysis software. A coding scheme was iteratively developed by the two authors with four top-level domains (Learning, Teaching, Mathematics, Miscellaneous) and several subcodes allowing for more nuanced coding. Separately from this coding scheme, declarative statements were highlighted.

Results

Thus far, we have identified three interesting findings. First, the interviews with the non-PhD and the PhD instructors took markedly different turns. For the non-PhD mathematicians, who had typically taught incoming undergraduate students, students' motivation was perhaps the most dominant issue. Both Aleph and Beth agreed that students require intrinsic motivation for true learning, which their students lacked. As a result, both had embraced efficiency in their teaching, for instance via course handouts containing most of the lectures' content. For the PhD mathematicians, motivation was hardly a topic of concern: They took intrinsic motivation for granted-perhaps because they were teaching mainly upper-division courses.

Second, the beliefs of the PhD mathematicians varied widely: Beliefs ranged from learning is solitary to learning is collaborative. Whereas Gimel believed that learning happened when one was working alone on a problem, Dalet took a more moderate approach and stated that learning could be both a solitary and a social activity. In opposition to Gimel's belief, Waw asserted that learning is social. The last PhD mathematician, Zayn, noted that he did not want to impose his own learning experiences onto others and that there are as many ways of learning as there are students. Thus, even among this small number of PhD mathematicians, beliefs varied widely, and no dominant view emerged.

Last, PhD mathematicians' own learning experiences aligned with how they spoke about their teaching. Gimel, Dalet, and Waw all taught in a way that reflected how they themselves had learned. Even Zayn, aware of the possible differences between his and his students' learning, realized during the interview: "[The course] is made in my image, now that I think about it. I went through all that trouble of saying I don't want them to learn the way I learned, but now that you're making me say it …" Thus, the PhD instructors' teaching mirrored their learning experiences.

Discussion

Our results lead us to ponder several issues. First, we wonder whether college mathematics is too broad a term. Perhaps a more fine-grained view that distinguishes between lower-and upperdivision courses might be beneficial for untangling college instructors' beliefs. Second, we note that instructors holding a wide variety of beliefs were able to win teaching awards at this institution. Lastly, the interview with Zayn demonstrated that one's teaching may be in-line with one's learning preferences even when one is aware of differences between one's own and others' learning experiences and wishes to be considerate of these differences.


Jean-Éric Pin

Xaro Soler-Escrivà

Languages and formations generated by D 4 and Q 8 *

We describe the two classes of languages recognized by the groups D 4 and Q 8 , respectively. Then we show that the formations of languages generated by these two classes are the same. We also prove that these two formations are closed under inverses of morphisms, which yields a language theoretic proof of the fact that the group formations generated by D 4 and Q 8 , respectively, are two equal varieties.

Most monoids and groups considered in this paper are finite. In particular, we use the term variety of groups for variety of finite groups. Similarly, all languages considered in this paper are regular languages and hence their syntactic monoid is finite.

Introduction

A nontrivial question is to describe the regular languages corresponding to well-studied families of finite groups. Only a few cases have been investigated in the literature: abelian groups [START_REF] Eilenberg | Automata, languages, and machines[END_REF], p-groups [START_REF] Eilenberg | Automata, languages, and machines[END_REF][START_REF] Weil | An extension of the Schützenberger product[END_REF][START_REF] Weil | Products of languages with counter[END_REF][START_REF] Weil | Closure of varieties of languages under products with counter[END_REF], nilpotent groups [START_REF] Eilenberg | Automata, languages, and machines[END_REF][START_REF] Thérien | Subword counting and nilpotent groups[END_REF], soluble groups [START_REF] Straubing | Families of recognizable sets corresponding to certain varieties of finite monoids[END_REF][START_REF] Weil | Products of languages with counter[END_REF] and supersoluble groups [START_REF] Carton | Languages Recognized by Finite Supersoluble Groups[END_REF]. More recently [START_REF] Ballester-Bolinches | Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited[END_REF], the authors addressed the following question: is it possible to obtain a reasonable description of the languages corresponding to a given formation of groups? Recall that a formation of groups is a class of finite groups closed under taking quotients and subdirect products.

This question was motivated by the importance of formations in finite group theory, notably in the development of a generalised Sylow's theory. The theory of formations was born with the seminal paper [START_REF] Gaschütz | Kennzeichnung gesättigter Formationen[END_REF] of [START_REF] Gaschütz | Kennzeichnung gesättigter Formationen[END_REF], where a broad extension of Sylow's and Hall's theories was presented. The new theory was not arithmetic, that is, based on the orders of subgroups.

It was concerned instead with group classes sharing certain properties, the so-called formations of groups, which have played a fundamental role in the study of groups since then [START_REF] Doerk | Finite soluble groups[END_REF][START_REF] Ballester-Bolinches | Classes of finite groups[END_REF].

In [START_REF] Ballester-Bolinches | Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited[END_REF], the authors extended Eilenberg's correspondence theorem between varieties of monoids and varieties of languages [START_REF] Eilenberg | Automata, languages, and machines[END_REF] to the setting of formations. More precisely, they spotted a bijective correspondence between formations of finite monoids and the so-called formations of languages. Using this "formation theorem" the authors not only recovered the previously mentioned results on nilpotent groups, soluble groups and supersoluble groups, but, relying on the local definition of a saturated formation [START_REF] Doerk | Finite soluble groups[END_REF], they exhibited new examples, like the class of groups having a Sylow tower [START_REF] Ballester-Bolinches | Languages associated with saturated formations of groups[END_REF].

The present paper focuses on the language interpretation of two results dealing with the dihedral group D 4 and the quaternion group Q 8 . The first result asserts that D 4 and Q 8 generate the same formation [START_REF] Doerk | Finite soluble groups[END_REF]Exercise 9,p. 344]. The second one states that this formation is a variety of groups, that is, is closed under taking subgroups. This latter result is actually an instance of a more general result, due to Neumann [START_REF] Neumann | A note on formations of finite nilpotent groups[END_REF], which states that any formation generated by a single nilpotent group is a variety (see [START_REF] Doerk | Finite soluble groups[END_REF]IV.1.16,p. 342] for an alternative proof).

The main result of this paper provides a purely language theoretic proof of these two results on D 4 and Q 8 . To do so, we first translate them in terms of languages: the formations of languages F 1 and F 2 associated to D 4 and Q 8 , respectively, are the same (first result) and they form a variety of languages (second result). The main difficulty in proving these results by pure language theoretic means is to establish the inclusion F 1 ⊆ F 2 . The lengthy proof of Theorem 5.1 should convince the reader that it is a nontrivial property.

Our proofs rely on a systematic use of the binomial coefficients of two words. This is not really a surprise, since binomial coefficients modulo p are the main tool for describing languages recognized by p-groups, and D 4 and Q 8 are 2-groups. In this paper, we present two explicit formulas with an algorithmic flavour. First, we discuss the behaviour of binomial coefficients under morphisms (Formula 3.4). Next, we show that a language of A * is recognized by a p-group if and only if it is a finite union of languages defined by linear algebraic constraints involving the binomial coefficients. Finally, we give an algorithm to obtain such a decomposition when the p-group is a group of unitriangular matrices over F p .

Our paper is organised as follows. In order to keep the paper selfcontained, prerequisites (Section 2) include formations and varieties, syntactic monoids and the Formation Theorem. Section 3 is devoted to binomial coefficients on words. We present in Section 4 various descriptions of the languages recognized by p-groups and the corresponding algorithms. Section 5 contains the proof of our main theorem.

Prerequisites

Formations and varieties

A formation of groups is a class of groups F satisfying the two conditions:

(1) any quotient of a group of F also belongs to F, (2) the subdirect product of any finite family of groups of F is also in F. Formations of finite algebras can be defined in the same way [START_REF] Shemetkov | Product of formations of algebraic systems[END_REF][START_REF] Skiba | Finite subformations of varieties of algebraic systems[END_REF][START_REF] Shemetkov | Formations of algebraic[END_REF]. In particular, a formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. If S is a set of finite monoids, the formation generated by S is the smallest formation containing S. It is also the set of quotients of subdirect products of members of S (see [5, A variety of groups is a class of groups V satisfying the three conditions:

(1) any subgroup of a group of V also belongs to V, (2) any quotient of a group of V also belongs to V,

the direct product of any finite family of groups of V is also in V. Varieties of monoids are defined in the same way. It follows from the definition that a formation of groups [monoids] is a variety if and only if it is closed under taking subgroups [submonoids]. Note that a formation is not necessarily a variety. For instance, the formation of groups generated by the alternating group A 5 is known to be the class of all direct products of copies of A 5 , which is not a variety [

Regular languages

A language is regular if it is representable by a regular expression. According to Kleene's theorem, a language is regular if and only if it is recognizable, that is, recognized by some finite automaton.

There is an equivalent definition in terms of monoids. A language L of A * is recognized by a monoid morphism ϕ : A * → M if there exists a subset P of the monoid M such that L = ϕ -1 (P ). By extension, L is said to be recognized by a monoid M if there exists a monoid morphism ϕ : A * → M that recognizes L. The equivalence mentioned above can now be stated as follows: a language is recognizable if and only if it is recognized by a finite monoid (see for instance [11, p. 15]).

Let L be a language of A * and let u be a word of A * . Then the language

u -1 L = {v | uv ∈ L}
is the left quotient of L by u.

The Nerode automaton of L is the deterministic automaton

A(L) = (Q, A, • , L, F ) where Q = {u -1 L | u ∈ A * }, F = {u -1 L | u ∈ L}
and the transition function is defined, for each a ∈ A, by the formula

(u -1 L)• a = a -1 (u -1 L) = (ua) -1 L.
Each state of A(L) is a left quotient of L by a word, and hence is a language of A * . The initial state is the language L, and the set of final states is the set of all left quotients of L by a word of L.

Proposition 2.1. A language L is recognizable if and only if the set {u -1 L | u ∈ A * } is finite. In this case, L is recognized by its Nerode automaton.

Syntactic monoids

Let L be a language and let x and y be words. The quotient x -1 Ly -1 of L by x and y is defined by the formula

x -1 Ly -1 = {u ∈ A * | xuy ∈ L}
The syntactic monoid of a language L of A * is the monoid obtained as the quotient of A * by the syntactic congruence of L, defined on A * as follows: u ∼ L v if and only if, for every x, y ∈ A * ,

xvy ∈ L ⇐⇒ xuy ∈ L
The natural morphism η : A * → A * /∼ L is the syntactic morphism of L. The syntactic monoid is the smallest monoid recognizing a language. In particular, a language is regular if and only if its syntactic monoid is finite.

A class of regular languages C associates with each finite alphabet A a set C(A * ) of regular languages of A * . It is closed under quotients if for each language L ∈ C(A * ) and for each pair of words (x, y) of A * , the language x -1 Ly -1 belongs to C.

The Formation Theorem

Just as formations of finite monoids extend the notion of a variety of finite monoids, formations of languages are more general than varieties of languages. Like varieties, formations are classes of regular languages closed under Boolean operations and quotients. But while varieties are closed under inverse of morphisms, formations of languages only enjoy a weak version of this property -Property (F 2 ) below -and thus comprise more general classes of languages than varieties.

The following definition was first given in [START_REF] Ballester-Bolinches | Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited[END_REF]. A formation of languages is a class of regular languages F satisfying the following conditions:

(F 1 ) for each alphabet A, F(A * ) is closed under Boolean operations and quotients, (F 2 ) if L is a language of F(B * ) and η : B * → M denotes its syntactic morphism, then for each monoid morphism α : A * → B * such that η • α is surjective, the language α -1 (L) belongs to F(A * ). Observe that a formation of languages is closed under inverse of surjective morphisms, but this condition is not equivalent to (F 2 ).

To each formation of monoids F, let us associate the class of languages F(F) defined as follows: for each alphabet A, F(F)(A * ) is the set of languages of A * whose syntactic monoid belongs to F.

Given a formation of languages F, let F(F) denote the formation of monoids generated by the syntactic monoids of the languages of F. The following statement is the main result of [START_REF] Ballester-Bolinches | Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited[END_REF].

Theorem 2.2 (Formation Theorem). The correspondences F → F(F) and F → F(F) are two mutually inverse, order preserving, bijections between formations of monoids and formations of languages.

Binomial coefficients on words

Binomial coefficients on words were first defined in [6, p. 238]. Useful references include [9, Chapter 6] and [START_REF] Reutenauer | Free Lie algebras[END_REF].

Definition of binomial coefficients on words

A word u = a 1 a 2 • • • a n (where a 1 , . . . , a n are letters) is a subword of a word v if v can be factored as v = v 0 a 1 v 1 • • • a n v n .
For instance, ab is a subword of cacbc. Given two words u and v, we denote by v u the number of distinct ways to write u as a subword of v.

More formally

, if u = a 1 a 2 • • • a n , then v u = Card{(v 0 , v 1 , . . . , v n ) | v 0 a 1 v 1 • • • a n v n = v}
Observe that if u is a letter a, then v a is simply the number of occurrences of the letter a in v, also denoted by |v| a . These binomial coefficients satisfy the following recursive formula, where u, v ∈ A * and a, b ∈ A:

           u 1 = 1 1 u = 0 if u = 1 va ub = v ub if a = b v ub + v u if a = b (3.1)
An alternative definition of the binomial coefficients is given below in Formula (3.3). We shall later use the following elementary result.

Binomial coefficients and morphisms

Let Z A be the ring of noncommutative polynomials with coefficients in Z and variables in A (see [START_REF] Lothaire | Combinatorics on words[END_REF]Chapter 6] or [START_REF] Reutenauer | Free Lie algebras[END_REF]). Given a polynomial P ∈ Z A and a word x, we let P, x denote the coefficient of x in P . Thus all but a finite number of these coefficients are null and P = x∈A * P, x x.

In this section, we study the behaviour of binomial coefficients under monoid morphisms. More precisely, given a monoid morphism ϕ : A * → B * and words u ∈ A * and x ∈ B * , we give a formula to compute ϕ(u)

x

.

The proof of this result relies on properties of the Magnus automorphism of the ring Z A . This automorphism µ A is defined, for each letter a ∈ A, by µ A (a) = 1 + a. Its inverse is defined by µ -1 A (a) = a -1. The following binomial identity [9, Formula 6.3.4]

for all u ∈ A * , µ A (u) = x∈A * u x x (3.3)
can be used to give an alternative definition of the binomial coefficients. If ϕ : A * → B * is a monoid morphism, then ϕ can be extended by linearity to a ring morphism from

Z A to Z B . Let γ : Z A → Z B be the ring morphism defined by γ = µ B • ϕ • µ -1
A . We are now ready to present the announced formula:

Proposition 3.2. Let ϕ : A * → B * be a morphism and let u ∈ A * and x ∈ B * . Then ϕ(u) x = |s| |x| u s γ(s), x (3.4)
Proof. Observing that µ -1 A (a) = a -1 for each letter a ∈ A, one gets

γ(a) = µ B (ϕ(a) -1) = µ B (ϕ(a)) -1 = x∈B * ϕ(a) x x -1 = x∈B + ϕ(a) x
x and thus γ(a), 1 = 0. It follows that γ(s), x = 0 if |x| < |s|. Furthermore, for each u ∈ A * , one gets on the one hand from (3.3)

µ B (ϕ(u)) = x∈B * ϕ(u) x
x and on the other hand, using (3.3),

γ(µ A (u)) = γ s∈A * u s s = s∈A * u s γ(s) = s∈A * x∈B * u s γ(s), x x Now since γ • µ A = µ B • ϕ,
(c) = a 2 b. First, γ = µ B • ϕ • µ -1
A is defined as follows:

γ(a) = µ B (ϕ(a -1)) = µ B (a -1) = µ B (a) -µ B (1) = (a + 1) -1 = a γ(b) = µ B (ϕ(b -1)) = µ B (ab -1) = (1 + a)(1 + b) -1 = a + b + ab γ(c) = µ B (ϕ(c -1)) = µ B (a 2 b -1) = µ B (a 2 b) -1 = (1 + a)(1 + a)(1 + b) -1 = 2a + aa + b + 2ab + aab

Languages recognized by p-groups

Let p be a prime number. A p-group is a group whose order is a power of p. A p-group language is a language whose syntactic monoid is a p-group.

Two descriptions of the p-group languages

The following result is credited to Eilenberg and Schützenberger in [START_REF] Eilenberg | Automata, languages, and machines[END_REF].

Proposition 4.1. A language of A * is a p-group language if and only if it is a Boolean combination of languages of the form

L(x, r, p) = {u ∈ A * | u x ≡ r mod p}, (4.5)
where 0 r < p and x ∈ A * .

We now give another characterization. A function f : A * → Z is said to be a linear combination of binomial coefficients if there exist c 1 , . . . , c n ∈ Z and x 1 , . . . , x n ∈ A * such that, for all u ∈ A * ,

f (u) = c 1 u x 1 + • • • + c n u x n (4.6) Since the function f (u) = c u 1
maps every word to the constant c, every constant function is a linear combination of binomial coefficients.

Proposition 4.2. A language of A * is a p-group language if and only if it is a finite union of languages of the form

L(f 1 , . . . , f r , p) = {u ∈ A * | f 1 (u) ≡ • • • ≡ f r (u) ≡ 0 mod p} (4.7)
where f 1 , . . . , f r are linear combinations of binomial coefficients.

Proof. Let G p be the Boolean algebra generated by the languages of the form L(x, r, p) and let S p be the set of languages that are finite unions of languages of the form L(f 1 , . . . , f r , p).

Step 1. S p is a Boolean algebra. First, S p is closed under union by definition. It is also closed under intersection since

L(f 1 , . . . , f r , p) ∩ L(g 1 , . . . , g s , p) = L(f 1 , . . . , f r , g 1 , . . . , g s , p). (4.8)
In particular,

L(f 1 , . . . , f r , p) = L(f 1 , p) ∩ • • • ∩ L(f r , p). (4.9)
It remains to show that S p is closed under complementation. Since S p is closed under union and intersection, it suffices to prove that the complement of each language of the form L(f, p), where f is a linear combination of binomial coefficients, belongs S p . Now

L(f, p) c = {u ∈ A * | f (u) ≡ 0 mod p} = c∈Fp\{0} {u ∈ A * | f (u) ≡ c mod p} = c∈Fp\{0} {u ∈ A * | (f -c)(u) ≡ 0 mod p}
It remains to observe that f -c is a linear combination of binomial coefficients to conclude.

Step 2: S p ⊆ G p . It suffices to show that every language of the form L(f, p) belongs to G p . Now if f is given by (4.6), one gets

L(f, p) = {(r 1 ,...,rn)|c 1 r 1 +•••+cnrn≡0 mod p} L(x 1 , r 1 , p) ∩ • • • ∩ L(x n , r n , p) (4
.10) and thus L(f, p) ∈ G p as required. Thus S p ⊆ G p .

Step 3: G p ⊆ S p . This immediately follows from the formula

L(x, r, p) = L(f, p) where f (u) = -r u 1 + u x .
Thus G p = S p and it now suffices to apply Proposition 4.1 to conclude the proof.

As explained in Section 2.2, one can compute the minimal automaton of a language of the form L(f 1 , . . . , f r , p) by computing its derivatives as follows:

u -1 L = {x ∈ A * | f 1 (ux) = f 2 (ux) = • • • = f n (ux) ≡ 0 mod p} .

An algorithm for p-group languages

Let p be a prime number and let U n (F p ) be the group of unitriangular1 n×nmatrices with coefficients in F p , the finite field of order p. Then U n (F p ) is a p-group and it is a well-known fact that every p-group is isomorphic to a subgroup of some U n (F p ), for a suitable choice of n. See for instance [13, p. 276, Corollary 5.48].

Let π : A → U n+1 (F p ) be a map2 and let G be the subgroup of U n+1 (F p ) generated by π(A). Then π extends to a surjective monoid morphism π : A * → G which maps every word a 1 • • • a k ∈ A * to the matrix π(a 1 ) • • • π(a k ). For 1 i < j n + 1, we let π i,j : A * → F p be the map defined, for all u ∈ A * , by

π i,j (u) = (π(u)) i,j (4.11)
By definition, a language K is recognized by π if there exists a subset S of G such that K = π -1 (S). According to Proposition 4.2, K is a finite union of languages of the form L(f 1 , . . . , f r , p). We now give an algorithm to obtain this representation explicitly.

Setting, for each s ∈ S, K s = π -1 (s), one gets

K = s∈S K s and K s = { u ∈ A * | for 1 i < j n + 1, π i,j (u) = s i,j }
It just remains to verify that the languages K s are of the form L(f 1 , . . . , f r , p). But this follows immediately from the following result:

Proposition 4.3. Each function π i,j is a linear combination of binomial coefficients.

Proof. Let θ : A → U n+1 (F p ) be the map defined by θ(a) = π(a) -1 for all a ∈ A. Then θ extends to a ring morphism θ : Z A → U n+1 (F p ) and for 1 i < j n + 1, the maps θ i,j : A * → F p are defined as in (4.11). Since θ(a) is a strictly triangular matrix for all a ∈ A, it follows that θ(x) = 0 for all words x of length > n. Note however that θ(x) is not in general equal to π(x) -1.

Let also µ : A * → Z A be the monoid morphism defined by µ(a) = 1+a for all a ∈ A. Thus µ is the restriction to A * of the Magnus automorphism introduced in Section 3.2. Since the formula θ(µ(a)

) = θ(1 + a) = 1 + θ(a) = π(a) holds for all a ∈ A, one has π = θ • µ. A * Z A U n+1 (F p ) π µ θ
It follows by (3.3) that

π(u) = θ(µ(u)) = θ x∈A * u x x = x∈A * u x θ(x) = |x| n u x θ(x)
and hence

π i,j (u) = |x| n θ i,j (x) u x (4.12)
which shows that π i,j is a linear combination of binomial coefficients.

An interesting special case occurs if the language is defined by constraints on the first row of the matrix, for instance for a language of the form

L = { u ∈ A * | π 1,2 (u) = • • • = π 1,n (u) = 0 }
Observing that L can also be written as

L = { u ∈ A * | (1, 0, . . . , 0)π(u) = (1, 0, . . . , 0) }
one can directly obtain a deterministic automaton for L by taking F n p as set of states, the state (0, . . . , 0) as initial and unique final state and by defining the transitions, for each (z 1 , . . . , z n ) ∈ F n p and each letter a, by setting

(z 1 , . . . , z n )• a = (z ′ 1 , . . . , z ′ n ), where (1, z 1 , . . . , z n )π(a) = (1, z ′ 1 , . . . , z ′ n ), (4.13)
that is,

z ′ 1 = π 1,2 (a) + z 1 , z ′ 2 = π 1,3 (a) + π 2,3 (a)z 1 + z 2 , z ′ 3 = π 1,4 (a) + π 2,4 (a)z 1 + π 3,4 (a)z 2 + z 3 , etc.
This algorithm is illustrated by the examples presented in Section 4.3.

Three examples

These examples will be used in Section 5. The languages of the first two examples were also considered by Thérien [START_REF] Thérien | Subword counting and nilpotent groups[END_REF].

1 0 0 0 1 0 0 0 1 a = 1 1 0 0 1 0 0 0 1 b = 1 0 0 0 1 1 0 0 1 ab = 1 0 1 0 1 0 0 0 1 ba = 1 1 0 0 1 1 0 0 1 aba = 1 0 1 0 1 1 0 0 1 bab = 1 1 1 0 1 0 0 0 1 abab = 1 0 1 0 1 0 0 0 1
Let π : A * → D 4 be the natural morphism and let

L 1 = {u ∈ A * | π 1,2 (u) = π 1,3 (u) = 0}.
To obtain a deterministic automaton for L 1 , we take F 2 2 as the set of states and define the transitions, for all (z 1 , z 2 ) ∈ F 2 2 , by setting

(z 1 , z 2 ) • a = (1 + z 1 , z 2 ) (z 1 , z 2 ) • b = (z 1 , z 1 + z 2 ) (4.14) (4.15)
The resulting automaton, which turns out to be minimal, is the following: Applying (4.12) with n = 2 one gets where the binomial coefficients are computed modulo 2. Thus the states of the minimal automaton of L 1 encode the possible values modulo 2 of these two binomial coefficients. Now, one can recover (4.14) and (4.15) by observing that, if

0, 0 1, 0 1, 1 0, 1
π 1,2 (u) = |x| 2 u x θ 1,2 (x) = u 1 θ 1,2 (1) + u a θ 1,2 (a) + u b θ 1,2 (b) + u aa θ 1,2 (aa) + u ab θ 1,2 (ab) + u ba θ 1,2 (ba) + u bb θ 1,2 (bb) = u a π 1,3 (u) = |x| 2 u x x 1,3 = u 1 θ 1,3 (1)
(0, 0)• u = z 1 , z 2 = u a , u ab then (0, 0)• ua = z 1 , z 2 • a = ua a , ua ab = u a + 1, u ab = (z 1 + 1, z 2 ) and (0, 0)• ub = (z 1 , z 2 ) • b = ub a , ub ab = u a , u ab + u a = (z 1 , z 1 + z 2 ) .
Example 4.2. The group D 4 is also generated by the two matrices

a = 1 1 0 0 1 1 0 0 1 and b = 1 1 1 0 1 0 0 0 1 A confluent rewriting system for this group is b 2 → 1, aba → b, ba 2 → a 2 b, bab → a 3 , a 4 → 1 and a 3 b → ba.
The group consists of the matrices

1 = 1 0 0 0 1 0 0 0 1 a = 1 1 0 0 1 1 0 0 1 b = 1 1 1 0 1 0 0 0 1 a 2 = 1 0 1 0 1 0 0 0 1 ab = 1 0 1 0 1 1 0 0 1 ba = 1 0 0 0 1 1 0 0 1 a 3 = 1 1 1 0 1 1 0 0 1 a 2 b = 1 1 0 0 1 0 0 0 1
Let π : A * → D 4 be the natural morphism and let

L 2 = {u ∈ A * | π 1,2 (u) = π 1,3 (u) = 0}.
To obtain a deterministic automaton for L 2 , we take F 2 2 as the set of states and define the for all (z 1 , z 2 ) ∈ F 2 2 , by setting

(z 1 , z 2 ) • a = (1 + z 1 , z 1 + z 2 ) (z 1 , z 2 ) • b = (1 + z 1 , 1 + z 2 ) (4.17) (4.18)
The resulting automaton, which turns out to be minimal, is the following:

0, 0 1, 0 0, 1 1, 1 a a, b
a =   1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1   b =   1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1   is isomorphic to Q 8 . A confluent rewriting system for this group is b 2 → a 2 , aba → b, ba 2 → a 2 b
, bab → a, a 4 → 1 and a 3 b → ba. The group consists of the matrices of the following form, where

ε 1 , ε 2 , ε 3 ∈ F 2 .   1 ε 1 ε 2 ε 3 0 1 0 ε 1 + ε 2 0 0 1 ε 2 0 0 0 1  
Let π : A * → Q 8 be the natural morphism and let

L 3 = {u ∈ A * | π 1,2 (u) = π 1,3 (u) = π 1,4 (u) = 0}.
To obtain a deterministic automaton for L 2 , we take F 3 2 as the set of states and define the transitions, for all (z 1 , z 2 , z 3 ) ∈ F 3 2 , by setting

(z 1 , z 2 , z 3 ) • a = (z 1 + 1, z 2 , z 1 + z 3 ) (z 1 , z 2 , z 3 ) • b = (z 1 , z 2 + 1, z 1 + z 2 + z 3 ) (4.20)

The varieties of languages V c,p

In this section, we revisit the congruences first introduced in [6, p. 240] and also studied in [START_REF] Thérien | Subword counting and nilpotent groups[END_REF]. Let c be a nonnegative integer. For each alphabet A, let ∼ p,c be the congruence on A * defined by u ∼ p,c v if and only if, for all words x such that 0 |x| c,

u x ≡ v x mod p
This congruence has finite index and the languages which are saturated for this congruence form a Boolean algebra V c,p (A * ), which is also the Boolean algebra generated by the languages L(x, r, p) for 0 r < p and |x| c.

Let us first show that the class V c,p is closed under inverses of morphisms. This result is due to Thérien [START_REF] Thérien | Classification of finite monoids: the language approach[END_REF], but for the convenience of the reader, we give a self-contain proof. This relies on the following result. We can now state: Proposition 4.5. Let ϕ : A * → B * be a morphism and L a language of V c,p (B * ). Then ϕ -1 (L) belongs to V c,p (A * ).

Proof. Let L be a language of V c,p (B * ). Let u ∈ ϕ -1 (L) and let v be a word such that u ∼ p,c v. Then ϕ(u) ∼ p,c ϕ(v) by Proposition 4.4, and since u ∈ L and L is saturated by ∼ p,c , we get ϕ(v) ∈ L, that is, v ∈ ϕ -1 (L). This proves that ϕ -1 (L) is saturated by ∼ p,c and therefore ϕ -1 (L) belongs to V c,p (A * ).

Proposition 4.6 ([18]). For each c, the class V c,p is a variety of languages.

Proposition 4.5 shows that the class V c,p is closed under inverses of morphisms. Furthermore, V c,p (A * ) is by definition a Boolean algebra, generated by the languages of the form L(x, r, p). We claim that it is closed under left quotient by a word u. Arguing on induction on the length of u, it suffices to consider the case where u is a letter a. Now, since left quotients commute with Boolean operations, it suffices to prove that any left quotient of the form a -1 L(x, r, p) belongs to V c,p (A * ). If x is the empty word, then L(x, r, p) is either empty or equal to A * and the result is trivial. Suppose that x is nonempty. Then, we get by (3.1): We are now ready to prove our main result. be the associated formation of languages. Let V = V 2,2 and let V be the associated group formation, which is actually a variety. For each alphabet A, V(A * ) is by definition the Boolean algebra generated by the languages L(x, r, 2) for 0 r < 2 and |x| 2. Proposition 4.6 shows that V is a variety. We shall prove successively the following properties:

a -1 L(x, r, p) = u ∈ A * | au x ≡ r mod p = {u ∈ A * | u x + u s ≡
(1) D 4 and Q 8 belong to V, and hence F 1 and F 2 are contained in V, (2) for each alphabet A, for 0 r < 2 and |x| 1, the language L(x, r, 2) belongs to F 1 (A * ) and to

F 2 (A * ), (3) V is contained in F 1 and hence V = F 1 , (4) F 1 is contained in F 2 .
In the sequel, the languages L 1 , L 2 and L 3 refer to the examples discussed in Section 4.3.

Step 1. The syntactic monoid of L 1 is equal to D 4 and that of L 3 is equal to Q 8 . Formula (4.16) shows that L 1 belongs to V({a, b} * ) and thus D 4 belongs to V. Moreover, Formula (4.22) shows that L 3 can be written as

L(a, 0, 2) ∩ L(b, 0, 2) ∩ i+j+k≡0 mod 2 (L(ab, i, 2) ∩ L(aa, j, 2) ∩ L(bb, k, 2))
and thus L 3 belongs to V({a, b} * ). It follows that Q 8 belongs to V.

Step 2. If x = 1, the result is trivial. If x = a, where a is a letter, the syntactic monoid of L(a, r, 2) is the cyclic group C 2 . Since C 2 is a quotient of both D 4 and Q 8 , it belongs to F 1 and to F 2 and thus L(a, r, 2) belongs to F 1 (A * ) and to F 2 (A * ).

Step 3. Let A be an alphabet. It suffices to prove that, for |x| 2 and r = 0 or r = 1, the language L(x, r, 2) belongs to F 1 (A * ). Let c(x) be the set of all letters occurring in x. In the minimal automaton of L(x, r, 2), every letter of A \ c(x) acts as the identity on the set of states. It follows that the languages L(x, r, 2) and the language u ∈ c(x) * | u x ≡ r mod 2 have the same syntactic monoid. Therefore, we may assume without loss of generality that A = {a, b}.

Suppose first that x = ab with a = b. It already follows from (2) that for |x| 1, L(x, r, 2) belongs to F 1 (A * ). Then the minimal automaton of L(ab, 0, 2) is obtained from the automaton of Figure 4.1 by taking (0, 0) and (1, 0) as final states. Indeed in this way the parameter z 2 = u ab will be equal to zero modulo 2. Thus the syntactic monoid of L(ab, 0, 2) is D 4 and since D 4 belongs to F 1 , the language L(ab, 0, 2) belongs to F 1 (A * ) and so does its complement L(ab, 1, 2).

Consider now the case x = aa. The automaton obtained from the automaton of the language L(aa, 0, 2) and its complement L(aa, 1, 2) belong to F 1 ({a, b} * ).

Since the languages L(bb, r, 2) and L(aa, r, 2) have the same syntactic monoid, we also have L(bb, r, 2) ∈ F 1 ({a, b} * ) for r = 0 and r = 1.

Step 4. We will show that some language L having D 4 as syntactic monoid belongs to F 2 . By the Formation Theorem, this will show that D 4 belongs to F 2 and hence that F 1 is contained in F 2 as required. We choose for L the language of Example 4.2:

L = ϕ -1 (

Proposition 3 . 1 .

 31 Let u ∈ {a, b} * . Then the following formula holds Let us prove (3.2) by induction on |u|. The result is trivial if |u| = 0. For the induction step, it suffices to prove the result for ua, the case ub being symmetrical.

  the polynomials µ B (ϕ(u)) and γ(µ A (u)) have the same coefficients, which gives (3.4). Example 3.1. To illustrate the use of (3.4), let us show how to compute ϕ(u) ab . Let A = {a, b, c}, B = {a, b} and let ϕ : A * → B * be the morphism defined by ϕ(a) = a, ϕ(b) = ab and ϕ

  We now need to compute the coefficients γ(s), ab for |s| 2. The non-zero coefficients are the following:γ(b), ab = 1 γ(c), ab = 2 γ(ab), ab = 1 γ(ac), ab = 1 γ(bb), ab = 1 γ(bc), ab = 1 γ(cb), ab = 2 γ(cc), ab =

Example 4 . 1 .

 41 The subgroup of U 3 (F 2 ) generated by the two matrices D 4 . A confluent rewriting system for this group is a 2 → 1, b 2 → 1 and baba → abab. The group consists of the matrices 1 =

Figure 4 . 1 :

 41 Figure 4.1: The minimal automaton of L 1 . The syntactic monoid of L 1 is the group D 4 presented by the relations a 2 = 1, b 2 = 1 and (ba) 2 = (ab) 2 . Its syntactic image is {1, b}.

L 1

 1 = u ∈ {a, b} * |

Figure 4 . 2 :

 42 Figure 4.2: The minimal automaton of L 2 .

Example 4 . 3 .

 43 The subgroup of U 4 (F 2 ) generated by the two matrices

Figure 4 . 3 : 1 , 2 (

 4312 Figure 4.3: The minimal automaton of L 3 .

Figure 4 . 4 :

 44 Figure 4.4: The Cayley graph of Q 8 .

Proposition 4 . 4 .

 44 Let ϕ : A * → B * be a morphism. Let u and v be two words of A * such that u ∼ p,c v. Then ϕ(u) ∼ p,c ϕ(v). Proof. If u ∼ p,c v, one has, for 0 |s| c, u s ≡ v s mod p. Therefore by (3.4) we obtain for |x| c, ), x ≡ 0 mod p Thus ϕ(u) ∼ p,c ϕ(v).

  r mod p} if x = as for some s {u ∈ A * | u x ≡ r mod p} otherwise = ∪ r 1 +r 2 ≡r mod p L(x, r 1 , p) ∩ L(s, r 2 , p) if x = as L(x, r, p) otherwise which proves the claim. A dual argument proves that V c,p (A * ) is closed under right quotient. Thus V c,p is a variety of languages. 5 The formation generated by D 4 and by Q 8

Figure 4 .

 4 2 by taking (0, 0) and (1, 0) as final states recognizes the languageK = u ∈ {a, b} * | of K is also D 4 and thus K ∈ F 1 (A * ). Now since L(aa, 0, 2) = (K ∩ L(b, 0, 2) ∩ L(ba, 0, 2)) ∪ (K ∩ L(b, 1, 2) ∩ L(ba, 1, 2)) ∪ (K c ∩ L(b, 0, 2) ∩ L(ba, 1, 2)) ∪ (K c ∩ L(b, 1, 2) ∩ L(ba, 0, 2))

2 1 κ 1

 211 1) = u ∈ {a, b} * | Let us now view D 4 as the group {1, a, b, a 2 , ab, ba, a 3 , a 2 b} presented by the group relations b 2 = 1, a 4 = 1 and a 3 b = ba and Q 8 as the group {1, a, b, a 2 , ab, ba, a 3 , a 2 b} presented by the group relations a 4 = 1, b 2 = a 2 and a 3 b = ba. Let B = {a, b} and C = {a, b, c}. Consider the following diagram, morphisms are defined byϕ(a) = a ϕ(b) = b α(a) = c α(b) = a ϕ 1 (a) = a ϕ 1 (b) = b ϕ 2 (a) = a ϕ 2 (b) = b ν 1 (a) = a 2 b ν 1 (b) = a ν 2 (a) = 1 ν 2 (b) = a and α 1 (a) = a α 1 (b) = b α 1 (c) = a 2 b α 2 (a) = a α 2 (b) = b α 2 (c) = (a) = a κ 1 (b) = b κ 1 (c) = a 2 b κ 2 (a) = a κ 2 (b) = b κ 2 (c) = 1 κ(a) = b κ(b) = 1 κ(c) = a

  II.2.2, p. 272] for group formations, [15, Chapter I, Theorem 2.2] and [8, Lemma 3.2] for general algebraic systems and [2, Proposition 1.4] for a self-contained proof for monoid formations).

  Theorem 5.1. groups D 4 and Q 8 generate the same formation and the associated formation of languages is the variety V 2,2 . Proof. Let F 1 [F 2 ] be the formation generated by D 4 [Q 8 ] and let F 1 [F 2 ]

An n × n-matrix is unitriangular if its diagonal coefficients are all equal to 1 and all its coefficients below the diagonal are equal to 0.

The switch from n to n + 1 will be justified later on.

Acknowledgements

We would like to thank Ramón Esteban-Romero, Adolfo Ballester-Bolinches and the anonymous referees for their useful comments and suggestions.

The first author is supported by Proyecto MTM2014-54707-C3-1-P from MINECO (Spain) and FEDER (European Union) and partially funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 670624) and by the DeLTA project (ANR-16-CE40-0007).

Applying (4.12) with n = 2 one gets 3 where the binomial coefficients are computed modulo 2. Thus the states of the minimal automaton of L 1 encode the possible values modulo 2 of these two linear combinations of binomial coefficients. Now, one can recover (4.17) and (4.18) by observing that, if

3 It is easy to make mistakes in this computation. Recall that in general θ(x) = π(x)-1.

Thus for instance θ(ba) = θ(b)θ(a) = 0 0 1 0 0 0 0 0 0 and π(ba)-1 = 0 0 0 0 0 1 0 0 0 , whence θ1,3(ba) = 1.

The syntactic monoid of L 3 is the group Q 8 presented by the group relations a 4 = 1, b 2 = a 2 and a 3 b = ba. Its syntactic image is {1}. The Cayley graph of this group is represented in Figure 4.4. As one can see, this is exactly the same automaton as in Figure 4.3, up to the following renaming of the states:

Note that ϕ 1 = ϕ 2 , but we keep two distinct names to preserve homogeneity of the notation. All these morphisms make the diagram commutative. Let

By construction, the languages R 1 , R b , R a 2 and R a 2 b are all recognized by Q 8 and hence belong to F 2 (B * ). Using the state renaming described in (4.23), one sees that R 1 , R b , R a 2 and R a 2 b are also accepted by the automaton represented in Figure 4.3 by taking as final state (0, 0, 0), (0, 1, 0), (0, 0, 1) and (0, 1, 1) respectively. Coming back to the interpretation of these states as linear combinations of binomial coefficients, as described in Example 4.3, one gets the following explicit descriptions:

We claim that

Indeed, Formula (3.4) leads to the following computations:

Finally R is the union of these four languages and hence .

It follows that

The syntactic monoid of R is D 4 and is syntactic morphism is κ.

Lemma 5.2. The language R belongs to F 2 (C * ).

Proof. For i = 1, 2, the morphism ϕ i •α i is equal to κ i and thus is surjective. By definition of a formation of languages, the languages α

The last step consists in computing α -1 (R).

Lemma 5.4. One has α -1 (R) = L and thus L belongs to F 2 ({a, b} * ).

Proof. Since ν

We claim that

Finally, Proposition 3.1 shows that when u a ≡ u b ≡ 0 mod 2, then u ab ≡ u ba ≡ 0 mod 2. It follows that α -1 (R) = L. This concludes the proof of Theorem 5.1.

Important remark. It is tempting to prove directly that the languages ν -1 1 (R 1 ), ν -1 2 (R 1 ), etc. belong to F 2 ({a, b} * ). However, the morphism ϕ 2 •ν 2 is not surjective and one cannot conclude directly.

Conclusion

We used language theory to prove that D 4 and Q 8 generate the same formation and that this formation is a variety of groups. Our project for the future would be to show, also by language theoretic means, that any formation generated by a single nilpotent group is a variety.