Guido Pusiol

Francois Bremond

Monique Thonnat

Unsupervised Discovery, Modeling and Analysis of long term Activities

This work proposes a complete framework for human activity discovery, modeling, and recognition using videos. The framework uses trajectory information as input and goes up to video interpretation. The work reduces the gap between low-level vision information and semantic interpretation, by building an intermediate layer composed of Primitive Events. The proposed representation for primitive events aims at capturing meaningful motions (actions) over the scene with the advantage of being learned in an unsupervised manner. We propose the use of Primitive Events as descriptors to discover, model, and recognize activities automatically. The activity discovery is performed using only real tracking data. Semantics are added to the discovered activities (e.g., "Preparing Meal", "Eating") and the recognition of activities is performed with new datasets.

Introduction

More than 2 billion people will turn over 65 year old by the year 2050. It is of crucial importance for the research community to help aging adults live independently for longer periods of time. The transition from their homes to new and unknown environments (i.e. an assisted living facility) add stressors that deteriorate theirs mind, memory and body. If we can keep the elders in their own homes over longer periods of time, they are in an environment that they know and trust so they can have a greater confidence leading to better quality of live.

The understanding of daily activities is the key to help solve the problem and is a topic that remains open. In the literature the computational approaches assume usually prior knowledge of the activities and the environment. This knowledge is used explicitly to model the activities in a supervised manner. In video surveillance the systems produce large quantities of data and it becomes almost impossible to continually monitor these data sources manually. It is of crucial importance to build computer systems capable of analyzing human behavior with minimal supervision.

Computer-based video applications need several processing levels, from low-level tasks of image processing to higher levels concerning semantic interpretation of the scene. Nowadays the reduction of the gap between low-level tasks up to video understanding is still a challenge.

This work addresses these problems by presenting a novel framework that links the basic visual information to the discovery and recognition of long term activities (e.g. "Eating") by constructing an intermediate layer of Primitive Events in a completely unsupervised way.

The intermediate layer aims at capturing the motion of the individual to perform basic tasks, using only minimal information (person position and dynamics). The use of small amounts of information allows the fast analysis of large amount of data. The advantage of using visual information is that it is captured using non-invasive sensors and enables to reduce the complexity of systems that use numerous sensors to enrich the observation data [START_REF] Zouba | Multisensor fusion for monitoring elderly activities at home[END_REF].

To automatically model the Primitive Events: First, the human actions are learned in an unsupervised way. Second, scene contextual information is learned capturing meaningful scene regions. Third, the primitive events are built by merging the actions and the scene information.

The composition of primitive events is very informative about the description of many activities. Thus, we search for particular sequences within the primitive event layer to discover interesting activities. The discovered activities are used to build generic activity models and the modeled activities are recognized in new unseen video datasets.

This paper is divided as follows: In the third section we explain how actions are learned, in the fourth section how the scene contextual information is obtained, in the fifth section how actions are abstracted to primitive events and how to combine the primitive events to discover and model activities, in the sixth section the activity recognition procedure is explained and in the seventh section we evaluate the approach in home-care applications.

Related Work

The advances made in the field of object tracking allow data-mining techniques to be applied to large video data. Recently particular attention has been focused on the object trajectory information over time to understand long term activities. Trajectory-based methods to analyze activity can be divided in two groups, supervised and unsupervised.

Typical supervised methods such as [START_REF] Haritaoglu | W4: Real-time surveillance of people and their activities[END_REF][START_REF] Laptev | Space-time interest points[END_REF][START_REF] Gong | Recognition of group activities using dynamic probabilistic networks[END_REF] can build activity models in a very accurate way. The problem is that they require big training datasets labeled manually.

The unsupervised methods include Neural Networks based approaches such as [START_REF] Johnson | Learning the distribution of object trajectories for event recognition[END_REF][START_REF] Hu | A system for learning statistical motion patterns[END_REF][START_REF] Owens | Application of the self-organizing map to trajectory classification[END_REF][START_REF] Khalid | Classifying spatiotemporal object trajectories using unsupervised learning of basis function coefficients[END_REF]. They can represent complex nonlinear relations between trajectory features in a low-dimensional structure. These networks can be trained sequentially and updated with new examples, but the complexity of the parametrization usually makes the networks grow and become useless after long periods of time.

Clustering approaches such as Hierarchical Methods [START_REF] Antonini | Trajectories clustering in ICA space: an application to automatic counting of pedestrians in video sequences[END_REF] allow multi-resolution activity modeling by changing the number of clusters, but the clustering quality depends on the way to decide when clusters should be merged or not. Adaptive methods [START_REF] Piciarelli | On-line trajectory clustering for anomalous events detection[END_REF], where the number of clusters adapts over time, make on-line modeling possible without the constraint of maintaining a training dataset. In these methods it is difficult to initialize a new cluster preventing outlier inclusion. Other methods [START_REF] Calderara | Detection of abnormal behaviors using a mixture of von mises distributions[END_REF][START_REF] Bobick | A state-based approach to the representation and recognition of gesture[END_REF] use dynamic programing based approaches to classify activitivities. These methods are effective when time ordering constraints hold.

Hidden Markov Model (HMM) based approaches such as [START_REF] Porikli | Learning object trajectory patterns by spectral clustering[END_REF] capture spatio-temporal relations in trajectory paths, allowing high-level analysis of an activity, which is suitable for detecting abnormalities. These methods require prior domain knowledge and their adaptability in time is poor.

Morris and Trivedi [START_REF] Morris | Learning and classification of trajectories in dynamic scenes: A general framework for live video analysis[END_REF] learn scene points of interest (POI) and model the activities between POIs with HMMs encoding trajectory points. This approach is suitable to detect abnormal activities and performs well when used in structured scenes (i.e. if the usual trajectory paths are well defined, such as on a highway). But the method requires activities to have time order constraints. Also [START_REF] Hamid | A novel sequence representation for unsupervised analysis of human activities[END_REF] merges the scene POIs and sensorial information. But the method requires a manual specification of the scene.

Most of the methods described above can be applied only in structured scenes (i.e. highway, traffic junction), and cannot really infer activity semantics. To solve these problems we propose an approach that is suitable to unstructured scenes and which is the first to combine local and global descriptors to recognize long term activities.

Actions

To understand activities, we propose first to learn the actions that compose them by cutting a video into meaningful action segments. Each segment aims at capturing a person's action such as "standing up". We mark the beginning and ending of a segment by detecting the person's change of state (motion/static). From a video datafile we obtain a sequence of action segments. At each segment we compute the person's main dynamics by clustering meaningful trajectories. Finally, we build Action descriptors that capture the global and local motion of a person in an action segment.

Global Position and Speed

We compute the person position at each frame by using a person tracker. The position is given to a linear Kalman-filter (K1). At each new frame the prediction of K1 is averaged with the new position observation (obs) obtaining a smoothed trajectory (pos):

pos f ramei = Avg(obs f ramei , K1(obs f ramei-1 ))
The speed of a person in a new frame, is computed by averaging the prediction of another Kalman-filter (K2) and the real speed observation (sobs) in the new frame:

speed f ramei = Avg(sobs f ramei , K2(speed f ramei-1 )

Action Segments and Local Dynamics

An Action Segment starts with a person's change of state and ends with the next change of state (motion/static). The changes of state are computed sequentially by thresholding the person's speed at each frame.

Local Dynamics are a set of short trajectories describing the motions in an action segment. To compute these trajectories, the algorithm starts by placing 500 KLT points [START_REF] Shi | Good features to track[END_REF] at the first frame of the action segment and tracks them [START_REF] Bouguet | Pyramidal implementation of the lucas kanade feature tracker description of the algorithm[END_REF] until the last frame. The resulting set of KLT trajectories is numerous and in long action segments noisy trajectories could appear. To filter the noise out we extract KLT trajectories where their start/end points are not far from the global position trajectory start/end points.

Several KLT trajectories could be describing the same motion; we cluster the KLT trajectories using Mean-Shift algorithm [START_REF] Georgescu | Mean shift based clustering in high dimensions: A texture classification example[END_REF] to obtain the main Local Dynamic trajectories. Mean-Shift is performed using the entry/exit points of the KLT trajectories to avoid the problem of clustering different trajectory lengths. The advantage of Mean-Shift is that it detects the number of clusters automatically, and filters out small clusters.

In Figure 1 displays a sequence of action segments with the computed Local Dynamics, it can be noticed how the small movements of the person are captured and that the resulting number of Local Dynamics is compact and descriptive.

Other descriptors have been tried (SIFT, SURF), they perform similarly to KLT but with much slower computational speed, while with KLT we process in real time.

Scene Context

In this approach, no information about the scene is known. We learn a scene model composed by scene regions in order to locate actions spatially. The type of regions we are interested in are those where the individual interacts with the scene objects (i.e.,"armchair"). The set of learned regions is called a topology and it is learned by clustering trajectory points.

Learning a Topology

To build a topology we use the Action posStart and Action posEnd spatial points from a sequence of actions. These points are features describing the locations where the changes of state occur and describe the locations of interaction with the scene. Let Action i be a sequence of actions. The set of InterestP oints used is:

InterestP oints = {Action i.posStart } ∪ {Action i.posEnd }
We perform K-Means clustering over InterestP oints. The number of clusters selected represents the level of abstraction of the topology, where lower numbers imply wider regions. Each cluster defines a Scene Region (SR). Finally, we denote T opology levelN = {SR 0 ...SR N }, where each SR i is labeled with a number for later use.

Scene Model

A scene model is composed by 3 topologies. They aim at describing coarse, intermediate and specific scene regions. Figure 2 displays 3 topologies composing the model of a scene that we use for experimentation.

Activities

In this section we explain how to combine actions and scene contextual information to discover and model activities. First, we build activity descriptors named Primitive Events that capture an Action's information over the scene. Second, we compute Primitive Event sequences of different levels of abstraction. Third, we combine the sequences to discover activities. And fourth, a discovered activity is modeled to be used by an activity recognition procedure.

Primitive Events

A Primitive Event (P E) is a descriptor that normalizes the global information of an Action using a scene model. Suppose an Action and a T opology then the P E resulting from Action is defined by its type as:

P E = (ST ART → EN D) (P E type).
where ST ART and EN D is the label of the nearest SR (Scene Region) of T opology to Action posStart/posEnd respectively.

ST ART = arg min

i (dist(Action posStart , SR i ))
The Action local descriptors are copied to the P E for later use.

(ST ART → EN D) Angles/Length = Action Angles/Length

Primitive Events Sequence

From a sequence of Actions, three Primitive Event sequences are computed. One for each T opology Level of a scene model. The motivation of having 3 levels of abstraction of P Es is that with the same set of descriptors, activities of different semantical abstraction levels can be discovered (e.g. "in the kitchen" and "at the kitchen sink").

Activity Discovery

Independently for each P E sequence described in the previous section, we extract particular subsequences that describe activity. We are interested in two types of subsequences, denoted SPOTTED and DISPLACED.

SPOTTED describes activity occurring within a single topology region (e.g. "Reading in the Armchair"). These are composed by P Es of the same type.

DISPLACED describes activity occurring between two topology regions (e.g. "from Bathroom to Table "). These are composed by a single P E.

Using regular expressions, a SP OT T EDA-A is a maximal subsequence of the P Es sequence of the type:

(A A) + (1)
A DISP LACEDA-B is a single P E of the type:

(A B), A = B (2)
The discovered SPOTTED and DISPLACED subsequences are presented to the user as displayed in Fig. 3. The user labels the subsequence that represents an interesting activity at any of the 3 abstraction levels. Adding a label to a subsequence SPOTTED or DISPLACED defines an ACTIVITY SPACE that contains the Primitive Events used to model the activity. An example of how an ACTIVITY SPACE is built is displayed in Fig. 4, where we use the 3 topologies displayed in Fig. 2 to represent a configuration of PE sequences. The example shows how the SPOTTED and DISPLACED subsequences are computed and examples of the ACTIVITY SPACEs defined by labeling as "Preparing Meal" SP OT T ED4-4 and as "In kitchen table" SP OT T ED6-6.

Activity Model

An Activity is modeled by 3 histograms (H 5 , H 10 , H 15 ) and a variable Activity Length .

Where H l captures the information of the PEs sequence of Level l contained in an AC-TIVITY SPACE. H l is an histogram of 2 dimensions. The first coordinate (globalf eature) is the type of a Primitive Event (S → E). The second coordinate (localf eature) is an angle value θ. The count is the accumulation of θ of the primitive events of type (S → E) appearing in the PEs sequence of Level l of the ACTIVITY SPACE.

H l (S → E, θ) = (S → E) i .Angles(θ) (3)
The Activity Length is the average length (S → E) Length of the Primitive Events appearing in the ACTIVITY SPACE.

Activity Recognition

For a new unseen video dataset, we aim at recognizing modeled activities in an unsupervised way. Suppose we have an Activity as well as the learned scene model used for modeling Activity. We are interested in finding a set of candidate activities that are similar to the modeled one. We explain the steps we use to find candidate activities in a new video: First, the sequence of actions is computed as described in Section 3.

Second, the Primitive Event sequences are computed, as described in Section 5.2. The difference is that this time we do not compute a new scene model, instead we use the learned scene model. This way, the PEs of the new video match spatially (PE type) with the PEs used for learning Activity.

Third, the activity discovery process is performed as described in Section 5.3. From the computed set of SPOTTED and DISPLACED subsequences, those that match the subsequence used for labeling Activity are selected. For example, in Figure 2 we label SP OT T ED4-4 to model "Preparing Meal". For the new video, all SP OT T ED4-4 appearing at Level 5 are selected. Fourth, the algorithm computes an ACTIVITY SPACE for each SPOTTED or DIS-PLACED selected in the previous step. From each ACTIVITY SPACE a candidate Activity is modeled as described in Section 5.4.

Fifth, because of the previous steps, a modeled Activity and a candidate Activity have a global spatial correspondence. But this does not ensures that both activities are the same (i.e. two different activities may take place at the same spatial location). To measure the similarity we compute score Length and score Histogram and we compare the values to thresholds T 1 and T 2. To obtain a binary recognition, an Activity is the same as Activity if the following statement is true:

score Length < T 1 ∧ score Histogram < T 2
Activity Similarity: We propose a distance that measures the similarity of all activity descriptors (local and global) by computing 2 scores between the model Activity and the candidate Activity . The score Length measures the similarity length of the local dynamics:

score Length = abs(Activity Length -Activity Length )
The score Histogram measures the similarity of the spatial position and local dynamic angles. This score is computed at the different levels of abstraction (capturing the subactivities similarity) by comparing the 3 histograms of Activity (H 5 , H 10 , H 15 ) with the 3 histograms of Activity . We experimented different similarity measures for multidimensional histograms and finally adopted Earth Movers Distance (EMD):

score Histogram = EM D(H i , H i )
Thresholds: The recognition thresholds T 1 and T 2 are learned using the information of the modeled activities. Let M odel 1 ... M odel i be of the same activity, we calculate the mean score Length and score Histogram of all combinations as well as their standard deviation σ 1 and σ 2 . Then T 1 and T 2 are defined as:

T 1 = Average(score Length ) + 2 * σ 1 T 2 = Average(score Histogram ) + 3 * σ 2

Experiments

For experimentation we use videos of 2 different scenes: HOME-CARE and HOSPI-TAL datasets. Each video contains a single person and are recorded using a monocular video camera (640 x 480 pixels of resolution). HOME-CARE contains 7 elderly people performing non-guided activities in an apartment (in total 24 hours of video). HOS-PITAL contains 4 videos of patients performing guided and non guided activities in a hospital room (3 hours of video). The last dataset is currently being used to study Alzheimer's disease symptoms and the protocol of the guided activities is described by Romdhane et al. [START_REF] Romdhane | Automatic video monitoring system for assessment of alzheimer's disease symptoms[END_REF].

From the discovered activities (i.e. Fig. 3) we label activities shared by most persons. They are selected using DISPLACED and SPOTTED subsequences, where the last ones are the most challenging because of possible activity confusions. For example, "Balance" and "Up/Down" are exercises for measuring the person's stability, both take place same location. The set of labeled activities is displayed in Tables 1,2.

Evaluation

The Activity Recognition method depends on the Activity Discovery method, therefore the evaluation of the first one reflects the quality of the discovery procedure.

We evaluate the activity recognition method using cross validation technique. The evaluation is performed recognizing activities in a test video by learning the scene and activity models from the remaining videos. For example, in HOME-CARE, to recognize activities of person G, we compute the scene and activity models using the videos of persons A,B,C,D,E,F. In total 6 experiments are performed (one for each test video). Performance measurements: For each dataset an activity ground truth (GT) is manually labeled. The GT describes the intervals of time when an activity begins and ends. The Activity Recognition method returns the intervals of time where an activity is recognized. Each recognized activity instance is compared with the GT and the following measurements False Recognition Time (FT): Percentage of time the activity is recognized while it is not occurring in the GT, over the time the activity is recognized. Results: Table 1 and Table 2 display the recognition results. In both datasets the method has a very good performance. The FP occurs when the motion of the person while doing different activities is similar and the FN because of the lack of motion. The FT occurs because a person stop an activity without changing of place (i.e. at the end of Eating stays still for a while). To illustrate the complexity of the recognized activities we display some results graphically in Fig. 5.

Activity

TP FP FN RT FT Balance 3 0 0 100% 1% Up/Down 3 0 0 100% 4% Reading at the table 10

Conclusions

We propose a method to discover and recognize long term activities loosely constrained, in unstructured scenes. The insight of this paper is that it is the first time a complete framework links from the pixel level to complex semantics ("Eating"), using global and local features. Other approaches use either local or global features and the type of activities recognized can be considered as actions (sitting down in a chair).

The contributions are summarized as: An algorithm to learn a scene context (Activity Model); a data structure that combines global and local descriptors (Primitive Events); a method to combine small tasks to discover activities automatically; a method to recognize activities in new datasets. The evaluation results show that it can be used to study activities in home care applications and to perform fast and reliable statistics that can help doctors to diagnose diseases such as Alzheimer. Our future work is going to be the the extension of the approach to perform on-line activity recognition.

Fig. 1 .

 1 Fig. 1. From left to right a sequence of action segments with the computed KLT trajectories (pink) and Local Dynamics (red) after Mean-Shift clustering.

3. 3

 3 Action Descriptors An Action is a descriptor that captures global and local information of the trajectories in an action segment: Globally: Action posStart and Action posEnd are the global person's position at the start/end frames of an action segment. Locally: Action Length is the average length of the Local Dynamic trajectories and Action Angles is an histogram of the directions of the Local Dynamic trajectories normalized to {0 • , 45 • , 90 • , 135 • , 180 • , 225 • , 270 • , 315 • }.

Fig. 2 .

 2 Fig. 2. Computed scene model corresponding to HOME-CARE dataset. From left to right the topologies of level 5, 10 and 15 are displayed. The labeled white dot represent the Scene Region center and the surrounding points the cluster members.

Fig. 3 .

 3 Fig. 3. Activity Discovery of 2 datasets: HOSPITAL (a) and HOME-CARE (b). The scene model used for (b) is displayed in Fig. 2. The colored segments correspond to DISPLACED and SPOTTED subsequences, where the same color is the same subsequence type. For example, SPOTTED(1-1) labeled at the abstraction level 5 (a) corresponds to activity of the person in the chair region. The displayed images are representative actions of the discovered activities.

Fig. 4 .

 4 Fig. 4. Example of Activity Discovery sequences. Each layer represents a PE sequence at a level of abstraction. The brackets show the computation of SPOTTED and DISPLACE subsequences, and the ACTIVITY SPACEs are defined by labeling a SPOTTED or DISPLACE subsequence.

  are extracted: True Positive (TP): Number of activity instances correctly recognized. False Positive (FP): Number of recognized instances not appearing in the GT. False Negative (FN): Number of instances appearing in the GT not recognized.. Recognition Time (RT): Percentage of time the activity is recognized, over the GT duration of the activity.

Fig. 5 .

 5 Fig. 5. Marked with (*) are the recognized segments (TP) of the activities: (1) "Balance" and "Up/Down"; (2) "Preparing Meal" and "Eating"; (3) "Reading at the Armchair". The activities are aligned in time. Not marked segments are other -different-activities occurring at the same spatial location not matching with the model. At the top, images representing characteristic actions of the activities. (A) is a False Negative due to lack of motion; (B) is an example of how local motion occurs at the "Preparing Meal" location, but there is no global position matching; (C) is a False positive due to similar motion and global position with the activity model.

Table 1 .

 1 Recognition results of the selected activities for HOSPITAL dataset.

			Activity	TP FP FN RT FT
			Eating	31 1 0 97% 7%
			Reading in the Armchair 24 4 0 92% 11%
			Preparing Meal	52 6 3 83% 6%
		1 1 95% 3%	Standing at Armchair	11 2 0 95% 5%
	Preparing Coffe	7 1 0 88% 5%	Sitting at Eating place	8 0 1 99% 2%
	At the Computer	6 1 0 91% 4%	Inside the bathroom	14 2 0 82% 7%
	Excercice 1	3 0 0 99% 2%	Armchair to Table	32 4 0 96% 1%
	Excercice 2	3 0 0 99% 1%	Armchair to Kitchen	15 1 0 98% 3%

Table 2 .

 2 Recognition results of the selected activities for HOME-CARE dataset.


Dinh The Luc

Soubeyran Antoine

Variable preference relations: existence of maximal elements

Keywords: Variable preference relation, improving path, maximal point, maximum principle. JEL Classification: C60, C62

We consider variable preference relations, called also reference dependent preference relations which are typical in the study of dynamic models in economic theories. We introduce a new concept of weak consistency, a generalization of acyclicity, as an immediate regret condition for variable preferences. The main result to establish is on an existence criterion for maximal elements of a space equipped with a weak consistent variable preference relation. It is expressed via preference completeness condition which is equivalent to existence of aspiration points. As applications we show that a number of results known in the recent literature on maximum principles on a space with or without topological structure can be obtained from the unifying approach of this paper.

Introduction

Preference is one of the fundamental concepts in economics and social sciences. It refers to a way to put available alternatives in a certain order according to their degree of utility or satisfaction and aims at answering the question "stay or move?". In equilibrium theory a static point of view is dominant, that is, one is looking for situations in which all agents prefer to stay rather then to move; while in innovation theory (Schumpeter 1943 [22]) a dynamic point of view is more important, that is, one is interested in knowing when the agents prefer moving from their current situations than staying there. These points of views are dual, once a preference is given. In the classical models preference relations are a priori determined and constant. They are transitive, and sometimes complete, often defined by utility functions, which greatly facilitate applications of mathematical tools. However, in the real world preference relations vary with the context (experience, characters, emotions, mental state, social links, embeddedness...) and have evolutionary nature, generating dynamic processes of actions. This is why models with changing preferences appeal much attention of researchers for several decades (see Basmann 1956 [4], Bleichbrodt 2007[5], Koszegi and Rabin 2006 [16], Tversky and Kahneman 1991 [START_REF] Tversky | Loss aversion in riskless choice: a reference-dependent model[END_REF] among many others).

In the present study we consider preference relations, not necessarily being transitive or complete, or given by utility functions, but varying from state to state. Our overall concern is the question under which circumstances maximal elements exist for a variable preference relation without transitivity. Maximal elements play a central role in many economic models, including global maximum of a utility function and Nash equilibrium of a noncooperative game or equilibrium of generalized game (Debreu 1959 [9]). Existence of maximal elements has been extensively studied for constant preference relations and literature on it is abundant. However, to our knowledge there exists few papers dealing with existence of maximal elements in the case of variable preference relations. Koszegi 2010 [START_REF] Koszegi | Utility from Anticipation and Personal Equilibrium, Economic Theory[END_REF], [START_REF] Koszegi | A model of reference-dependent preferences[END_REF] [START_REF] Koszegi | A model of reference-dependent preferences[END_REF] and some others study a personal equilibrium of consumption in which a particular variable preference is used via variable utility functions (see also [START_REF] Gul | The revealed preference implications of reference dependent preferenes[END_REF] [START_REF] Gul | The revealed preference implications of reference dependent preferenes[END_REF]); Soubeyran 2009 [START_REF] Soubeyran | Variational rationality, individual stability and change: a theory of worthwhile and ambidextry behaviors[END_REF] considers a worthwhile to change function, which generates a variable preference relation to model a number of problems of social science such as habits, routines, behavioral traps etc. in terms of maximal elements. To establish existence criteria for maximal elements of a variable preference relation we exploit the concept of order-completeness borrowed from vector optimization which can be interpreted as a hypothesis on aspiration points recently introduced in the theory of change (see Soubeyran 2009 [23]) or Brezis-Browder's inductivity hypothesis [START_REF] Brezis | A general principle on ordered sets in nonlinear functional analysis[END_REF] [START_REF] Brezis | A general principle on ordered sets in nonlinear functional analysis[END_REF]) when the preference relation is determined by a utility function. Similar to the case of multi-criteria optimization given in Luc [START_REF] Luc | Theory of Vector Optimization[END_REF] and recently in Flores-Bazan et al. [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] the method of order-complete sets is very useful in unifying results on existence of maximal elements in various contexts, including (a) existence of maximal elements in social choice theory with non-transitive and incomplete preferences (Bergstrom 1975 [6] for acyclic relations, Tian and Zhou 1993 [START_REF] Tian | Transfer continuities, generalizations of the Weierstrass theorem and maximum theorerm: a full characterization[END_REF] for transfer upper continuous preferences, Zuanon 2009 [32] for weakly tc-upper semicontinuous acyclic relations, Alcantud 2002 [START_REF] Alcantud | Characterization of the existence of maximal elements of acyclic relation[END_REF] for upper continuous preference on " "-upper compact sets, Andrikopoulos and Zacharias 2009 [START_REF] Andrikopoulos | Characterization of the existence of R-maximal elements of consistent binary relations[END_REF] for consistent upper tc-S-semicontinuous preferences); (b) maximum ordering principles in partially ordered spaces without compactness [START_REF] Altman | A generalization of the Brezis-Browder principle on ordered sets[END_REF] [START_REF] Altman | A generalization of the Brezis-Browder principle on ordered sets[END_REF], Turinici 1984 [28], Szaz 2007 [START_REF] Szaz | An improved Altman type generalization of the Brezis-Browder ordering principle[END_REF], Zeidler 1986 [30], Zhu and Li 2007 [START_REF] Zhu | Generalization of ordering principles and applications[END_REF] in the context of partial ordering); and (c) efficiency conditions in the framework of generalized multi-criteria optimization (Flores-Bazan et al. 2008 [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF]).

The paper is structured as follows. In Section 2 basic concepts and elementary properties of variable preference relations are given. We introduce the concept of weak consistency which generalizes well-known concepts of acyclicity, path consistency and consistency by Suzumura. We establish equivalence between acyclicity of a variable preference relation and its transitive closure being a partial order. Section 3 deals with a central concept of our paper: exante maximal points and their existence. We prove a criterion for existence of maximal elements in a space equipped with a variable preference relation. It is based on the idea of order-complete sets from vector optimization and the weak consistency introduced in Section 2. Section 4 is devoted to applications of the main existence result of Section 3 to several models of recent literature. Related notions such as ex-ante ideal elements, ex-post ideal elements and expost maximal elements are also briefly discussed. Applications of our approach to behavioral traps and equilibrium theory will be addressed in a forthcoming work.

Basic concepts and elementary properties

Let X be a set of actions or choices of an agent or a group of agents, or a set of states of an economic system. A preference relation on X that evolves from state to state is called a variable preference relation. Reference dependent, variable, dynamic and evolutionary preferences have the same meaning in our context. They all describe a system in which at a given state x ∈ X an agent has a set of criteria that forms a preference relation allowing him to choose a new state to move to. At this new state, again, he has a set of criteria that may differ from the one he has had at x and generates a new preference relation for the next move. Examples of variable preference relations are found in multicriteria decision making, consumption theory, psychology and other areas of social sciences (see Giraud 2004 [12]), Kahneman and Tversky 1979 [14], Lindblom 1959 [17] and references given therein).

In this section we develop fundamental concepts related to variable preference relations and some of their elementary properties. Let us first recall basic definitions for constant preference relations. A preference relation P on X is a relation linking pairs of elements of X, that is, P is a binary relation on X and defined by a subset R of the product space X × X as follows: for x, y ∈ X, one has xP y (we say y is preferred to x) if and only if (x, y) ∈ R. A preference relation P is said to be reflexive if xP x for all x ∈ X; it is irreflexive if xP x is true for no x ∈ X; it is antisymmetric if xP y and yP x implies x = y; and it is transitive if xP y and yP z imply xP z. A reflexive, antisymmetric and transitive preference relation is called a partial order. Sometimes irreflexive partial orders are also considered. Reflexivity of a preference relation occurs in a system when an agent is happy with his position, that is, according to his criteria, the current position is acceptable for his next move. Irreflexivity occurs when an agent is unhappy with his position and definitely wishes to move to another position at the next step. If a preference relation P is not reflexive, mathematically one may generate an associated reflexive preference relation by adding the diagonal of the space X × X to the set R. Similarly, if a preference relation is not irreflexive, by deleting the diagonal from the set R one obtains an irreflexive preference relation. In general, assuming a preference relation reflexive is widely accepted by researchers and practitioners. Moreover, if P is reflexive, a strict preference relation associated to P is defined to be an irreflexive preference relation P smaller than P in the sense that xP y implies xP y (or equivalently the set R defining P is a subset of R). When a reflexive preference relation P is transitive, one defines equivalent classes on X as follows: (

) 2
Here is a link between these strict preference relations. Throughout this section we will assume that the space X is equipped with a variable preference relation {" ≤ x " : x ∈ X} and that " ≤ x " is reflexive. Given an element x ∈ X, the following sets are of particular interest at the individual or collective levels, ex-ante (before moving) and ex-post ( after moving) levels : i) The "ex-ante" dominant and dominated sets at x,

S + (x) = {y ∈ X : x ≤ x y} S -(x) = {y ∈ X : y ≤ x x}.
Thus, S + (x) can be viewed as the set of states that dominates x by the criteria at x, that is, states to which from his own point of view, being at the state x, an agent or a group of agents wish to move; while S -(x) contains all states that are dominated by x, that is, states to which, being at x and according to their own judgements, an agent or a group of agents do not want to move.

ii) The "ex-post" dominant and dominated sets at x,

F + (x) = {y ∈ X : x ≤ y y} F -(x) = {y ∈ X : y ≤ y x}.
The terminology "ex-post" refers to the fact that, starting from x and moving to y, an agent or a group of agents prefer staying at y ∈ F + (x) to coming back to x, this being judged by criteria at y after having moved from x.

From now on we shall fix the strict preference relation " < x " associated to " ≤ x " by relation [START_REF] Alcantud | Characterization of the existence of maximal elements of acyclic relation[END_REF], that is x < x y if and only if x ≤ x y and y ≤ y x (the negation of y ≤ y x). The ex-ante strict dominant and dominated sets at x are defined in a similar manner with strict preference relation " < x " instead of " ≤ x " and denoted respectively S > (x) and S < (x):

S > (x) = {y ∈ X : x < x y} S < (x) = {y ∈ X : y < x x}.
Likewise, the sets of ex-post strict dominant and dominated sets at x are denoted F > (x) and F < (x). An evident relationship between elements of the above sets is the following:

y ∈ S + (x) (resp. y ∈ S > (x)) if and only if x ∈ F -(y) (resp. x ∈ F < (y)); (3) y ∈ S -(x) (resp. y ∈ S < (x)) if and only if x ∈ F + (y) (resp. x ∈ F > (y)).
In general, when the variable preference relation is not constant, the ex-ante and ex-post dominant/ dominated sets are distinct. Let us now define paths of acceptable changes or improving paths. Given two states x and y of X, an upward path from x to y is a finite sequence of elements x 1 , ...,

x n ∈ X such that x = x 1 ≤ x1 x 2 ≤ x2 x 3 ..... ≤ xn-2 x n-1 ≤ xn-1 x n = y. ( 4
)
An upward path from x to y means that an agent, being at the state x k , prefers to move to x k+1 along this path. A downward path from x to y is a sequence of elements x 1 , ..., x n ∈ X such that

x = x 1 ≤ x 2 x 2 ≤ x 3 x 3 ... ≤ x n-1 x n-1 ≤ x n x n = y.
This means that once arrived at the state x k an agent prefers staying at this state to coming back to the previous state x k-1 . When n = 2 the above paths are called direct; otherwise they are called indirect (because y is preferred to x not directly, but through intermediate states

x i for 1 < i < n).
Since variable preference relations are generally not transitive, upward paths and downward paths provide a way to obtain transitive preference relations on X. Let us define the upper transitive closure and the lower transitive closure of the variable preference relation " ≤ x " as follows:

• x ≤ u y if and only if there is an upward path from x to y;

• x ≤ y if and only if there is a downward path from x to y.

Upper and lower transitive closures being transitive preference relations, we may define equivalent classes on X. The induced preference relations are denoted respectively " u " and " ". In view of Lemma 1 there is no distinction between strict preference relations by ( 1) and ( 2), so we denote them by " ≺ u " and " ≺ ". Below is a relationship between a variable preference relation and its transitive closure.

Proposition 2 Assume that x ≤ u y is given by [START_REF] Basmann | A theory of demand with variable consumer preferences[END_REF]. Then

y ≤ u x implies x k+1 ≤ x k+1 x k for some k ∈ {1, ..., n -1} y = x implies x k+1 = x k for some k ∈ {1, ..., n -1}. Moreover, if [x k+1 ] u [x k ] for some k ∈ {1, ..., n -1}, then [y] u [x].
Proof. The first relation follows from the fact that if x k+1 ≤ x k+1 x k for all k ∈ {1, ..., n -1}, then y ≤ u x which contradicts the hypothesis. The second relation is evident. For the last assertion, suppose to the contrary that

[y] u [x].
There is an upward path from y to x. Then the upward path

x k+1 ≤ x k+1 x k+2 ... ≤ xn y ≤ u x = x 1 ≤ x1 x 1 ... ≤ x k-1 x k yields x k+1 ≤ u x k , implying [x k+1 ] = [x k ]
. This contradiction completes the proof. Related to paths of acceptable changes we present some basic concepts which are already known for constant preference relations. The concept of weak consistency seems to be new and generalizes acyclicity.

Transitivity. The variable preference relation " ≤ x " is said to be transitive if x ≤ u y implies x ≤ x y for all x, y ∈ X. In other words if two states are joint by an indirect path, then they can be joint by a direct path. Antisymmetry. The variable preference relation " ≤ x " is said to be antisymmetric if x ≤ x y and y ≤ y x imply x = y. We notice that given x 0 ∈ X, as an individual preference relation on X, (≤ x0 ) need not be transitive or antisymmetric even if the variable preference relation is such.

Acyclicity. The variable preference relation " ≤ x " is said to be acyclic if for every upward path from x to itself, all intermediate elements coincide with x. Equivalently, if there are finite numbers of elements x 1 , ..., x n ∈ X such that an agent moves from x = x 1 to x n through x 2 , ..., x n-1 and at least one intermediate state x k is different from x, then x = x n . Path consistency. The variable preference relation " ≤ x " is said to be path consistent if for every x ∈ X there is no upward path from

x to itself x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n =
x in which at least one intermediate preference is strict in the sense of (1), that is x k+1 ≤ x k+1 x k for some k ∈ {1, ..., n -1}. We see that acyclicity implies path consistency, but the converse is not true in general. Actually, acyclicity can be seen as path consistency when ( 2) is used instead of (1): there is no path from x to itself, say

x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n = x
such that at least one intermediate preference is strict in the sense that x k = x k+1 for some k. Suzumura's consistency. The variable preference relation " ≤ x " is said to be consistent in Suzumura's sense if whenever there is an upward path from x to y = x, either there is a direct path from x to y, or no direct path from y to x. In other words, for every x, y ∈ X, x = y and x 1 , ...,

x n ∈ X such that x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1
x n = y (the agent wants to pass indirectly from x to y), then either x ≤ x y (the agent wants to pass directly from x to y) or y ≤ y x (the agent does not want to pass directly from y to x). This definition is an adaptation of consistency by Bossert and Suzumura 2007 [START_REF] Bossert | Social norms and rationality of choice[END_REF] for usual preference relations. Weak consistency.

The variable preference relation " ≤ x " is said to be weak consistent if whenever there is a direct path from x to y and an indirect path from y to x, there is a direct path from y to x. In other words,

x ≤ x y ≤ y x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n = x implies y ≤ y x.
This condition can be interpreted as an immediate "regret" of having moved from x to y after a long (indirect) way from y to x. Here is a relationship between the notions of consistency given above.

Proposition 3 Every path consistent preference relation is Suzumura consistent, and every Suzumura consistent preference relation is weak

consistent. Proof. Assume that x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n = y. If y ≤ y x, then we have an upward path x 1 , ..., x n ∈ X such that x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n = y ≤ y x. If the variable preference relation is path consistent, then all intermediate relations are not strict in the sense of (1), that is, x k+1 ≤ x k+1 x k for k = 1, ..., n -1 and x ≤ x y. By this, the preference relation is Suzumura consistent. Now, let x ≤ x y ≤ y = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ x n-1 x n =
x be an upward path from x to itself. If the variable preference relation is Suzumura consistent, then one must have y ≤ y x, because x ≤ x y is not true.

It can be seen without difficulty that the converse of Proposition 3 is not true. The utility of acyclicity and consistency is seen from the next result, see also Proposition 6.

Proposition 4 Given a variable preference relation " ≤ x " on X, the induced preference relation " u " is a partial order on equivalent classes of X. Moreover, the transitive closure " ≤ u " is a partial order on X if and only if the variable preference relation is acyclic.

Proof. We first observe that being reflexive and transitive the transitive closure " ≤ u " defines equivalent classes on X. And the induced preference relation " u " is antisymmetric on equivalent classes, hence it is a partial order. Now assume that " ≤ u " is a partial order on X.

Let x = x 1 ≤ x 1 x 2 ≤ x 2 x 3 ..... ≤ x n-2 x n-1 ≤ xn-1 x n =
x be an upward path from x to itself. Then for k = 2, ..., n -1 one has x ≤ u x k and x k ≤ u x. By antisymmetry x = x k . Hence the variable preference relation is acyclic. Conversely, assume the preference relation is acyclic. Then x ≤ u y and y ≤ u x imply x ≤ u y ≤ u x. Consequently, y is an intermediate element of an upward path from x to itself. By acyclicity, all intermediate elements of that path coincide with x, in particular y = x. The proof is complete.

In a recent work by Andrikopoulos and Zacharias 2009 [START_REF] Andrikopoulos | Characterization of the existence of R-maximal elements of consistent binary relations[END_REF] the authors have found some sufficient conditions under which a constant preference relation P is acyclic (respectively Suzumura's consistent) if and only if there there exist two functions G :

X → [0, 1] and C : X × X → [0, 1] such that for every x, y ∈ X, xP y ⇐⇒ G(y) -G(x) > C(x, y) (respectively either G(y) -G(x) > C(x, y) or G(y) = G(x).)

Ex-ante maximal points

As we have already seen in the previous sections, variable preference relations describe economies in movement. A move from a state x to a state y can be targeted or not. In a move without target the agent has no goal to orient his action, he just wishes to improve his situation as in muddling through process. By changing from x ∈ X to y ∈ X, he passes from an ex-ante perception to an ex-post perception. In an ex-ante perception the agent starts from doing x and chooses to carry out a new action y, driven by an ex-ante motivation to change: x ≤ x y. From his initial point of view at x, he prefers moving to y than staying at x. In an ex-post perception, once arrived at y (having done the new action y ), the agent compares, from his new point of view at y, his new situation y with the old one x. Either he does not regret, from his new point of view at y, to be at y because x ≤ y y, or he regrets his move because y ≤ y x. Given x ∈ X, the ex-ante dominant set S + (x) is exactly the ex-ante preference change set at x. It contains all possible improving moves from x according to the point of view at x. In a targeted move, a state x being given, the agent starts from a state y with y ≤ y x. He sets x as his target to move to. The ex-post dominated set F -(x) at x contains all states y at which the agent does not want to stay and from which he wishes to move to the target x. Then an important question is to know whether a state exists such that once arrived there, the agent stops his move, finding no better place to go. This is the concept of maximal points that we are going to develop now.

Throughout this section the space X is equipped with a reflexive variable preference relation " ≤ x ". Below we define maximal points with respect to the variable preference relation, and give some characterizations via dominant and dominated sets. Then we introduce the concept of preference-complete sets borrowed from vector optimization (Luc 1989 [START_REF] Luc | An existence theorem in vector optimization[END_REF], see also Application 2, Section 4) and prove a general criterion for existence of maximal points. A particular variant of this criterion is proposed when the space is equipped with a topology which turns to be quite useful in applications.

Definition 5 An element x * is said to be an ex-ante maximal point of X if there is no y ∈ X such that the direct path from x * to y is possible, but the direct path from y to x * is impossible. It is said to be an ex-ante strict maximal point or a rest point of X if there is no point y = x * such that the direct path from x * to y is possible, that is, x * ≤ x * y.

As we have already discussed the terminology "ex-ante" comes from the fact that an agent or a group of agents starts from x * and considers, before moving, the opportunity to move from x * to some y. It is clear that every rest point (cycle of order zero) is an ex-ante maximal element (cycle of order zero and one). The converse is generally not true except for the case of antisymmetric preference relations. The definition above is applied to any binary relation on X. Let P be a reflexive and transitive preference relation on X, we say that x * is P -maximal point of X if there is no y ∈ X such that xP y and not yP x; and it is a P -rest point if there is no y ∈ X such that xP y and x = y. In other words, x * is a P-maximal point if there is no y ∈ X such that x * P y, and it is a P-rest point if there is no y ∈ X such that x * P y in the sense of (1) and ( 2) respectively. Again we notice that when P is antisymmetric, these two notions coincide (see Duggan 2007 [10]).

Proposition 6 Let x * be given. The following assertions hold. 1) There is equivalence between: (a) x * is an ex-ante maximal element;

(b) S > (x * ) = ∅; (c) x * ∈ ∪ y∈X F < (y).
2) There is equivalence between:

(d) x * is a rest point; (e) S + (x * ) = {x * }; (f ) x * ∈ ∪ y∈X,y =x * F -(y). 3) If x * is ≤ u -rest point,
y = x 1 ≤ x 1 x 2 ...x n-1 ≤ x n-1 x n = x * .
Completing this path by adding x * ≤ x * y to the first term of the path we obtain an upward path from x * to itself:

x * ≤ x * y = x 1 ≤ x1 x 2 ...x n-1 ≤ xn-1 x n = x * .
In view of weak consistency we deduce y ≤ y x * . This proves that x * is an ex-ante maximal element of X.

Maximal elements include the case of personal habits, organizational routines, behavioral traps, personal equilibrium ( Koszegi and Rabin 2006 [START_REF] Koszegi | A model of reference-dependent preferences[END_REF]), Nash equilibrium, generalized Nash equilibrium, evolutionary stable equilibrium (Thomas 1985 [26]), norms, conventions (Young 1993 [29]), innovation frontier, Pareto solutions, bargaining solutions, focal points (Schelling 1960 [21]), ideal points, cycling points (Duggan 2007 [10]), recurrent points or recurrent sets, drops, top cycle set, untrapped set etc.

Existence of maximal elements without topology. We recall some standard definitions related to monotone nets. A net {y α } α∈I , where I is a directed index set, is said to be (upper) strictly increasing if y α < u y β whenever α < β. It is said to be dominated by a net {x β } β∈I if I ⊆ I and for every α ∈ I there is some β ∈ I such that y α ≤ u x β . Given a point a ∈ X, the upper section of X at a is denoted by S(a) := {x ∈ X : a ≤ u x}. Definition 7 A subset of X is said to be preference-complete (or P-complete for short) if it has no covering of the type {X \ S(x α ) : α ∈ I} with {x α } α a strictly increasing net in that subset.

Notice that in the above definition no topological structure is necessary on the space X. However, in most applications X is topological and we shall explicitly assume it when necessary. Sufficient conditions for P-completeness will be given later when we consider specific models (see . For the moment let us present the main existence result of this section. There are several ways to prove it under acceptance of Zorn's lemma (see also Flores-Bazan et al. 2008 [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] for constant preferences). The approach below is much inspired by the one of Luc 1989 [18] for efficient points in vector optimization and based on Proposition 6.

Theorem 8 Assume that the variable preference relation " ≤ x " on X is weakly consistent. Then X has ex-ante maximal points if and only if there is some point a ∈ X such that the upper section of X at a is P-complete.

The proof of this theorem as well as its applications (Section 4) are given in Appendix. We underline that the merit of the above theorem resides in the sufficient part: if X admits a P-complete section, then it has maximal element. It is to note that when X has maximal elements, it may have sections that are not P-complete. As we'll see later the concept of P-completeness is quite convenient in deriving different kinds of existing conditions for maximal elements when the preference relation is constant. Let us interpret P-completeness in terms of aspiration states. Given a strictly increasing net {x α , α ∈ I} in X, the family {X \ S(x α ) : α ∈ I} is a covering of X if and only if for each x ∈ X there is some index α ∈ I such that x ∈ S(x α ). Consequently, Pcompleteness of X is equivalent to the fact that for each strictly increasing net as above, there is a point x * ∈ X such that x * ∈ S(x α ) for all α ∈ I. This point x * can be considered as an "aspiration state" of the net. When I is the set of the natural numbers N, a sequence {x n : n = 1, 2, ...} with x n ≤ x n x n+1 is called an improving process. If x * is an aspiration state of the improving process {x n : n = 1, 2, ...}, then it belongs to the ex-ante dominant set S + (x n ) for all n ≥ 1. Hence along the improving process if the agent discovers an aspiration state, he moves directly to it without passing through the whole process, economizing time, money and energy. If in addition the aspiration state x * is ex-ante strict maximal, then it is called an aspiration driven equilibrium of the improving process. Thus, an aspiration driven equilibrium of the process {x n : n = 1, 2, ...} satisfies two conditions: a) x * ∈ S + (x n ) for all n ≥ 1, and b) S + (x * ) = {x * }. In the framework of partially ordered spaces the hypothesis of existence of aspiration points has already been given by Brezis and Browder under the name of inductivity hypothesis in their famous principle in Brezis and Browder 1976 [START_REF] Brezis | A general principle on ordered sets in nonlinear functional analysis[END_REF]. In fact, suppose that X is equipped with a partial order (≤). The inductivity hypothesis is satisfied if a) every improving sequence {x n } n≥1 with x n ≤ x n+1 has an upper bound, and b) there exists a potential function Φ : X → R which is bounded above and increasing, i.e. Φ = sup {Φ(x), x ∈ X} < +∞ and x ≤ y =⇒ Φ(y) ≥ Φ(x). It is clear that for every x ∈ X, there exists x * ∈ X such that x ≤ x * and Φ(y) = Φ(x * ) for all y such that x * ≤ y. Moreover, if in addition x ≤ y and x = y imply Φ(x) < Φ(y), then S + (x * ) = {x * }, which means that x * is an aspiration driven equilibrium. The value Φ(x * ) is called an improving goal payoff.

Before presenting a variant of Theorem 8 under the presence of a topology on X, we stress once more the fact that Theorem 8 is valid for reflexive variable preferences which are not necessarily transitive, or antisymmetric. This is an important point because it allows to study a lot of anomalies raised by Kahneman and Tversky 1979 in [START_REF] Kahneman | Prospect theory: an analysis of decision under risk[END_REF] and their followers (we'll address this issue and applications of variable preferences in another study).

Existence of maximal elements with topology. Throughout this part we assume that X is a topological space. The concepts of upper closedness and upper compactness to introduce below are essential for the study of variable preferences in topological spaces.

Definition 9

The variable preference relation " ≤ x " on X is said to be upper closed if for every convergent strictly increasing net {y α } α∈I with limit y, either S > (y) is empty, or there is some z ∈ X such that y α < u z for all α ∈ I. The set X is said to be upper compact if every upper strictly increasing net in X is dominated by a strictly increasing net that has a convergent subnet.

Compact sets are first evident examples of upper compact sets because every net of a compact set has a convergent subnet. The converse is not true in general. For instance the set of all negative numbers together with 0 is not compact, but upper compact, the topology and order being usual. As to upper closedness of variable preference relations, an immediate sufficient condition is that all upper sections are closed, that is, for every a ∈ X, the upper section {x ∈ X : a ≤ u x} is closed. In fact, let {x i } i∈I be a strictly increasing net converging to x in X. Let i, j ∈ I with j > i. Then x j ≤ u x and x i < u x j . Hence x i < u x for every i ∈ I. A condition more general then closedness of upper sections is known as transfer upper continuity and will be given in Application 3. Of course, an upper closed preference relation may have non-closed upper sections. For instance the preference relation " " on [0, 2] defined by x x y if and only if x ≤ y and y = 1, is upper closed, but the upper section at 0 is not closed. The result below is an useful consequence of Theorem 8. It makes use of upper compactness and upper closedness, and is quite ready for applications.

Corollary 10 Let X be a topological space. Assume that the variable preference relation " ≤ x " is upper closed, weakly consistent and that for some point a ∈ X, the section S(a) is upper compact. Then S(a) admits a P-complete subsection and consequently X has ex-ante maximal points.

Proof. Suppose to the contrary that S(a) has no P-complete subsections. In particular there is a strictly increasing net {x i } i∈I in S(a) such that the family {S(a) \ S(x i ) : i ∈ I} is a covering of S(a). In view of upper compactness hypothesis there is a strictly increasing net in S(a) that dominates {x i } i∈I in S(a) and admits a convergent subnet {y j } j∈J . It can be seen that the family {S(a) \ S(y j ) : j ∈ J} is a covering of S(a) too. Let y be the limit of the subnet {y j } j∈J . Since S(a) has no P-complete subsection, y cannot be a maximal point, hence there is some z ∈ S(a) such that y j ≤ u z for all j ∈ J because the variable preference " ≤ x " is upper closed. Consequently, S(a) is not covered by the family {S(a) \ S(y j ) : j ∈ J}, a contradiction. The proof is complete.

Applications

In this section we show that a number of existing results for maximal elements from the literature under divers contexts can be derived from the criterion we've developed in the previous section. By our knowledge all of them deal with constant preference relations and many of them assume that the relation under consideration is a partial order.

Application 1: Generalization of the Brezis-Browder principle. Let X be a nonempty set (for us an action set) with a reflexive and transitive preference relation " ≤ ". It is said to be countably inductive (CIO set for short) if every nondecreasing sequence has an upper bound. Let W be a nonempty set (for us a payoff or value set) with a partial order " ". It is said to be totally ordered upper separable if for every totally ordered nonempty set M ⊆ W , there is a nondecreasing sequence {v n } n≥1 ⊂ M such that every v ∈ M is dominated by some v n . In term of satisficing, this assumption means that one can always build a satisficing scale in any totally ordered nonempty payoff (utility) subspace M ⊂ W.

Corollary 11 Assume that X is a CIO set, Y is a totally ordered upper separable set and G : X → Y is a nondecreasing map. Then for every x 0 ∈ X, there is some x * with x 0 ≤ x * such that G(x) = G(x * ) for all x with x * ≤ x.

The above corollary is the ordering principle by Zhu and Li in a recent work [START_REF] Zhu | Generalization of ordering principles and applications[END_REF] which generalizes the well-known Brezis-Browder principle as we have already said.

Application 2: General multicriteria optimization. Let X be a nonempty set equipped with a partial order " ≤ " and let A be a nonempty subset of X. In multicriteria or vector optimization we are interested in finding maximal elements of A when X is a vector space and in most cases the order is generated by a convex cone. In a recent work [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] Flores-Bazan et al. have studied this question in a space without linear structure, generalizing the existing results in vector optimization. Their main result (Theorem 3.1) states that the set A has a maximal element if it is order-totally-complete in the sense that it has no covering of the form X \ S + (x), x ∈ D, where D is a totally ordered subset of A. It is clear that an order-totally-complete set is P-complete. Hence Theorem 3.1 (see also Theorem 4.3) of Flores-Bazan et al. 2008 [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF]) can be derived from Theorem 8 by a direct application. Note that in [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] (Definition 2.2) the authors use nonincreasing nets to define order-complete sets, while in [START_REF] Luc | Theory of Vector Optimization[END_REF] and in Definition 5, we use strictly increasing nets. Working with equivalent classes one may prove that a P-complete set is order-totally-complete, and so, applying Theorem 3.1 of [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] to the transitive closure " ≤ u " is also an alternative proof of Theorem 8. Of course, the weak consistency must be then taken into account to produce a desired result. The proof given in Appendix details the argument by Zorn's lemma as exploited in [START_REF] Flores-Bazan | Characterizing zfficizncy without linear structure: a unifying approach[END_REF] and [START_REF] Luc | An existence theorem in vector optimization[END_REF].

Application 3: Transfer upper-continuous preference relation. We say that a variable preference relation " ≤ x " is transfer upper-continuous if x < x y implies the existence of some y ∈ X and a neighborhood U (x) in X such that z < z y for all z ∈ U (x). The concept of transfer upper-continuity was introduced in Tian and Zhou 1995 [START_REF] Tian | Transfer continuities, generalizations of the Weierstrass theorem and maximum theorerm: a full characterization[END_REF] for a fixed binary relation on X which generalizes upper continuity of binary relations ( x < x y implies z < z y for all z in some neighborhood of x). The corollary below is a generalization of the main results of Bergstrom 1975 [6] and Tian and Zhou 1995 [START_REF] Tian | Transfer continuities, generalizations of the Weierstrass theorem and maximum theorerm: a full characterization[END_REF].

Corollary 12 Assume that X is a compact set of a topological space and that the variable preference relation is acyclic and transfer continuous. Then X is upper compact, and the variable preference relation is upper closed and weakly consistent. Consequently X has ex-ante maximal points.

It is to note that transfer continuity implies upper closedness. The converse is not true in general. Here is an example. Consider X = A ∪ B where

A = { 2 n -1 2 n : n = 1, 2, ...} B = { 3 n -1 3 n : n = 1, 2, ...} ∪ {1; 2}.
A binary relation " " is defined as follows: x y if and only if x ≤ y (in the usual sense) and x, y ∈ B. It is evident that the preference relation generated by that binary relation is upper closed. It is not transfer continuous under the usual topology of the real numbers because 2 > 1 and in every neighborhood of 1 there are elements of A which are not comparable with 2.

Application 4: Weakly tc-upper semicontinuous preference relation.

We say that the variable preference relation " ≤ x " is weakly tc-upper semicontinuous if it is acyclic and for x < x y there is an open set U (x, y) satisfying (i) x ∈ U (x, y) and y ∈ U (x, y);

(ii) F -(z) ⊆ U (x, y) for every z ∈ U (x, y); (iii) U (x, y) ⊂ U (z, w) and inclusion is strict if x < x y, z < z w and y ∈ U (z, w).
The concept of weak tc-upper semicontinuity was introduced in Zuanon 2009 [START_REF] Zuanon | A note on maximal elements for acyclic binary relations on compact topological spaces[END_REF] for binary relations which generalizes transfer continuity. As we shall see in the proof of the next corollary that this generalization can be reduced to the case of transfer continuity.

Corollary 13 Assume that X is a compact set of a topological space and that the variable preference relation " ≤ x " is weakly tc-upper semicontinuous. Then X has ex-ante maximal points.

Application 5: Upper continuous preference relation on " ≺ "-upper compact sets. Let Y be a topological space and let " " be a partial order on Y . We say that Y is " "-upper compact if every covering of Y by a family of open free disposal sets of Y admits a finite subcovering. Recall that a set D ⊆ Y is free disposal if x ≺ y, y ∈ D implies x ∈ D. The terminology "free disposal" is taken from Debreu 1959 [START_REF] Debreu | Theory of Value: an axiomatic analysis of economic equilibrium[END_REF] in which the usual Pareto partial order of R n is considered. Free disposal sets are called lower comprehensive in Alcantud 2002 [START_REF] Alcantud | Characterization of the existence of maximal elements of acyclic relation[END_REF]. The concept of " ≺ "-upper compactness was introduced in Alcantud 2002 [START_REF] Alcantud | Characterization of the existence of maximal elements of acyclic relation[END_REF] which generalizes compactness.

Corollary 14 Assume that the variable preference relation " ≤ x " is acyclic such that the sets {y ∈ X : x < u y}, x ∈ X are open and that X is " < u "-upper compact. Then X is P-complete and consequently it has ex-ante maximal points.

The above corollary is essentially Theorem 3 of Alcantud 2002 [START_REF] Alcantud | Characterization of the existence of maximal elements of acyclic relation[END_REF] when the preference relation is constant.

Application 6: Consistent upper tc-S-semicontinuous preference relation. A (Suzumura) consistent preference relation " ≤ x " is called upper tc-S-semicontinuous (in the sense of Andrikopoulos and Zacharias 2009 [START_REF] Andrikopoulos | Characterization of the existence of R-maximal elements of consistent binary relations[END_REF]) if for every x ∈ X, the set {y ∈ X : y < u x} is open. We derive the following result of Andrikopoulos and Zacharias 2009 [START_REF] Andrikopoulos | Characterization of the existence of R-maximal elements of consistent binary relations[END_REF] (Theorem 3).

Corollary 15 Assume that the variable preference relation " ≤ x " is consistent (in Suzumura's sense) and upper tc-S-semicontinuous and X is " < u "-upper compact. Then X has ex-ante maximal points.

Application 7: Maximum principle. Another generalization of the Brezis-Browder principle given by Szaz 2007 [START_REF] Szaz | An improved Altman type generalization of the Brezis-Browder ordering principle[END_REF] (see also a less general version by Altman 1982 [2], Turinici 1984 [START_REF] Turinici | A generalization of Altman's ordering principle[END_REF] and Zeidler 1986 [30]) can also be obtained from Theorem 8. Let X be a nonempty set with a binary relation " ≤ ". Let ∆ be a function on X × X with values in the extended real line R. Define a function γ on X by

γ(x) = sup x≤y ∆(x, y).
Here is a generalization of the maximum principle in Szaz 2007 [START_REF] Szaz | An improved Altman type generalization of the Brezis-Browder ordering principle[END_REF].

Corollary 16 Assume that there are a point a ∈ X and a function ∆ : X×X → R such that (i) γ is decreasing;

(ii) γ(x) > -∞ for all x ∈ X;

(iii) γ(a) < ∞;

(iv) Every increasing sequence {x n } n≥1 in S(a) with x 1 = a is bounded above and lim inf n→∞ ∆(x n , x n+1 ) ≤ α;

(v) ∆(x, y) > α for every x < y.

Then X has an ex-ante maximal point in S(a).

Related notions

Other concepts of maximality with respect to a variable preference relation is related to the best action of an agent or a group of agents when he compares ex-ante, before moving from x * , the current action x * with all others, and the concept of ex-post maximal points which is a counterpart of ex-ante maximal points when downward paths are used.

Definition 17 An element x * is said to be an ex-ante ideal point if y ≤ x * x * for every y ∈ X.

It is clear that an ideal point is maximal, but the converse is not true in general. For instance the segment joining the points (1, 0) and (0, 1) in R 2 has no ideal points, but every point of it is maximal, the preference relation being the usual Pareto order in R 2 . An ideal point, if it exists, is not necessarily unique, except for the case when the preference relation is antisymmetric. It is easy to check that x * ∈ X is ex-ante ideal if and only if

S -(x * ) = X, or equivalently x * ∈ ∩ y∈X F + (y). (5)
We shall make use of the last condition to find sufficient conditions of existence of ex-ante ideal points. To this end let us recall the following result of [START_REF] Luc | Existence of solutions in variational relation problems without convexity[END_REF]: If a family {A i : i ∈ I} of subsets of a compact space is intersectionally closed and has the finite intersection property, then the intersection of its members is nonempty. The family {A i : i ∈ I} of nonempty subsets in a topological space is intersectionally closed if cl(∩ i∈I A i ) = ∩ i∈I cl(A i ); and it has the finite intersection property if the intersection of any finite number of elements of the family is nonempty. Several sufficient conditions for intersectional closedness are found in Luc et al. 2010 [START_REF] Luc | Existence of solutions in variational relation problems without convexity[END_REF].

Theorem 18 Assume that X is compact and the following conditions hold:

(i) For a finite set D ⊆ X, there is a point x D ∈ X such that y ≤ x D x D for every y ∈ D;

(ii) The family {F + (x) : x ∈ X} is intersectionally closed.

Then X has ex-ante ideal points.

Proof. In view of (i), the family {F + (x) : x ∈ X} has the finite intersection property, and by (ii) it is intersectionally closed. According to the finite intersection principle, there is some x * ∈ F + (x) for all x ∈ X. By [START_REF] Bleichbrodt | Reference-dependent utility with shifting reference points and incomplete preferences[END_REF] x * is an ex-ante ideal point of X.

As a consequence of the above theorem we deduce an existence condition for quasi-convex preference relations. Assume that X is a convex subset of a vector space. The variable preference relation "

≤ x " is quasi-convex if x ≤ y y and x ≤ z z imply x ≤ u u for every u = ty + (1 -t)z, t ∈ [0, 1].
Corollary 19 Assume that X is a nonempty convex and compact subset of a Banach space, and that the following conditions hold: (i) For every x ∈ X the set F + (x) is convex and closed;

(ii) The variable preference relation is quasi-convex.

Then X has ex-ante ideal points.

Proof. Since the set F + (x), x ∈ X are closed, Condition (ii) of Theorem 18 is satisfied. Moreover, (ii) implies that the family {F + (x) : x ∈ X} has the finite intersection property. It remains to apply Theorem 18 to complete the proof.

Ex-post maximal points. In this last part we consider a move from x to y, not from the "ex-ante" point of view, before moving from the origin x, but, "ex-post", after moving, from the point of view of the arrival point y. As before, " ≤ x " is a reflexive variable preference relation on X.

Definition 20 An element x * is said to be ex-post maximal if there is no other element y ∈ X such that x * < y y ; and it is said to be ex-post ideal if y ≤ y x * for all y ∈ X.

Then, an element x * is ex-post maximal if there is no point y such that, once arrived at y, from the point of view of y, starting from x * , an agent or a group of agents strictly prefer to stay at y than to come back to x * . An element x * is ex-post ideal if once moved to any y, and based on the judgement at y, an agent or a group of agents regret that move. We can see also that x * is ex-post maximal if and only if

F > (x * ) = ∅; ( 6
)
and it is ex-post ideal if and only if

F -(x * ) = X, (7)
or equivalently

x * ∈ ∩ x∈X S + (x). (8)
Notice also that when the variable preference relation " ≤ x " is constant in the sense that for every x, y ∈ X the preference relations " ≤ x " and " ≤ y " coincide, there is no distinction between ex-ante and ex-post maximalities. It is important to note that existence conditions for ex-post maximal points follow the same pattern of ex-ante maximal points. Therefore we do not further go into details of these conditions. Let us give just one counterpart of Theorem 18 as an example.

Theorem 21 Assume that X is a nonempty compact set and that the following conditions hold.

(i) For a finite set D ⊆ X, there is a point x D ∈ X such that x ≤ x x D for every x ∈ D;

(ii) The family {S + (x) : x ∈ X} is intersectionally closed.

Then X has ex-post ideal points.

Proof. Apply the argument of Theorem 18 and the relation [START_REF] Brezis | A general principle on ordered sets in nonlinear functional analysis[END_REF]. We close up this section by observing that under (ii) of Theorem 21 Condition (i) is equivalent to the existence of ex-post ideal points. Moreover, the compactness of X can be relaxed by compactness of an upper section S + (x 0 ) for some x 0 ∈ X. This is because when the family of S + (x) is intersectionally closed, then the family of S + (x) ∩ S + (x 0 ) is also intersectionally closed. It is clear that the above observation is valid for ex-ante ideal points as well. increasing net, which means that it belongs to P. Indeed, let x and y be two distinct elements of a 0 . There are two indices λ and λ from Λ such that x is an element of a λ and y is an element of a λ . Since D is a chain, we may assume that a λ is included in a λ . Thus, both x and y are elements of the strictly increasing net a λ , and so either x < u y or y < u x. This proves that a 0 is strictly increasing. Moreover, it is evident that a a 0 for all a ∈ D. Hence a 0 is an upper bound of X. According to Zorn's lemma there is a maximal element a * , say a * = {x i } i∈I ∈ P. We claim that the family {x ∈ S(a) : x i ≤ u x} i∈I is a covering of S(a). Indeed, since a * is maximal, there is no x ∈ S(a) such that x i ≤ u x for all i ∈ I. In other words, for every x ∈ S(a) there is some i ∈ I such that x i ≤ u x. Thus, the above mention family of sets is a covering of S(a) and we arrive at a contradiction with the P-completeness hypothesis. By this X has an ex-ante maximal point. We now proceed to the case where the upper transitive closure " ≤ u " is not a partial order. Consider the induced preference " u " on equivalent classes of X. It is a partial order on [X]. Denote by Ŝ(a) the section of [X] at [a] with respect to the induced order:

Ŝ(a) = {[x] ∈ [X] : [a] u [x]}.
It can be seen that [x] ∈ Ŝ(a) if and only if x ∈ S(a). We show that Ŝ(a) is P-complete with respect to the induced preference. Indeed, let {[x α ]} α∈I be a strictly increasing net with respect to the induced preference in Ŝ(a). Then {x α } α∈I is a strictly increasing net with respect to " ≤ u " in S(a). If Ŝ(a) were covered by the family {[X] \ Ŝ(x α ) : α ∈ I}, then for every x ∈ S(a) there would exist some

i ∈ I such that [x] ∈ [X] \ Ŝ(x i ). Then [x] ∈ Ŝ(x i ), which implies [x i ] u [x]
, and so x i ≤ u x. We deduce that x ∈ X \ S(x i ), which contradicts the P-completeness of S(a). Now we apply the first part of the proof to the induced preference to obtain a u -maximal element [x * ] of [X]. Since the variable preference is weakly consistent, in view of Proposition 6, x * is an ex-ante maximal element of X as well. The proof is complete.

Proof of Corollary 11. By working on equivalent classes if necessary one may assume without loss of generality that X is partially ordered. Now define a new preference relation " ¡ " by x ¡ x if and only if either x = x , or x < x and G(x) < G(x ). It is clear that this relation is a partial order. Consider a section S(x 0 ) of X at x 0 (with respect to the new relation). We show that it is P-complete. Indeed, let {x i } i∈I be a strictly increasing net in S(x 0 ), then {G(x i )} i∈I is an increasing net, hence, by choosing a totally ordered subnet if necessary, we may assume it is a totally ordered set in Y . By hypothesis there is a sequence {x n } n≥1 from the above net such that for each i ∈ I there is some n(i) with G(x i ) ≤ G(x n(i) ). We may assume {G(x n )} n≥0 is increasing, which implies that {x n } n≥0 is strictly increasing with respect to " ≤ ". By the CIO assumption, there is an upper bound of {x n } n≥0 which is also an upper bound for the net {x i } i∈I . Then that upper bound belongs to S(x i ) for all i ∈ I and the family {X \ S(x i ) : i ∈ I} does not cover S(x 0 ). Thus S(x 0 ) is P-complete.

According to Theorem 8 there is a maximal point x in S(x 0 ) for the relation " ¡ " which is a point we look for.

Proof of Corollary 12. Since the preference relation is acyclic, it is weakly consistent. The upper compactness of X follows from its compactness as we have already mentioned. It remains to prove the upper closedness of the variable preference relation. Let {x α } α∈I be a convergent strictly increasing net in X. Let x be its limit. If S > (x) is empty, we are done. If S > (x) is nonempty, say x < x y for some y ∈ X, then by transfer continuity, there is some y ∈ X and a neighborhood U such that z < z y , z ∈ U. In particular, there is α 0 ∈ I such that x α < xα y for all α > α 0 . This implies that x α < u y for all α ∈ I (choose β ∈ I with α < β and α 0 < β, then x α < u x β and x β < x β y implying x α < u y ). Thus, all the assumptions of Corollary 10 hold and so X has maximal points.

Proof of Corollary 13. Let {U λ : λ ∈ Λ} denote the family of all open sets in the definition of weak tc-upper semicontinuity. Define a new preference relation on X by x ≺ y if there is some U λ such that x ∈ U λ and y ∈ U λ . It is easy to see that this new preference added by the diagonal of the product space X × X is a partial order on X which is upper continuous, hence transfer continuous. In view of Corollary 12 the set X has a maximal element with respect to that new order. It is clear that that maximal element is also maximal with respect to the given variable preference relation on X.

Proof of Corollary 14. Since the preference is acyclic, it is weakly consistent as well. We prove that X is P-complete. Indeed, let {x α } α∈I be a strictly increasing net in X. Consider the family A α := X \ cl(S(x α )), α ∈ I, where cl(.) stands for the closure. It is clear that every member of this family is open and free disposal. If the family {X \ S(x α ) : α ∈ I} were a covering of X, then so would the family {A α : α ∈ I} be, and by hypothesis, it would admit a finite covering, say A αi , i = 1, ..., n. Take any β > α i , i = 1, ..., n. Then x α i < u x β , i = 1, ..., n. This implies that x β ∈ A α i for all i = 1, ..., n and we arrive at a contradiction. It remains to apply Theorem 8 to complete the proof.

Proof of Corollary 15. Denote by [X] the space of equivalent classes of elements of X with respect to the preference " ≤ x "; and equip [X] with the quotient topology and the quotient preference. It is known that when the preference is consistent, the quotient preference is a partial order. Moreover, with respect to the quotient topology, the quotient preference is continuous and [X] is upper compact. Applying Corollary 13 to the space [X] we obtain a maximal class, say [x]. Then any element of this class is an ex-ante maximal point of X with respect to the given variable preference (see Proposition 6).

Proof of Corollary 16. Consider the section S(a). It follows from (i)-(iii) that γ(x) is finite for every x ∈ S(a). It is also clear from (i) that the transitive closure of the binary relation " ≤ " is path consistent. Thus, by working on equivalent classes if necessary one may assume without loss of generality that the transitive closure " ≤ u " is a partial order on S(a). We show that the section S(a) is P-complete. Indeed, let {x i } i∈I be a strictly increasing net in S(a), then one may choose an increasing (with respect to the transitive closure) sequence {x i(n) } n≥1 from that net such that lim n→∞ γ(x i(n) ) = lim i γ(x i ).

Here the limit may take the value -∞. Then for every i ∈ I, there is some i(n) such that x i ≤ x i(n) . Indeed, given i ∈ I, by the choice of the sequence {x i(n) } n≥1 there is i(n) such that γ(x i(n) ≤ γ(x i ). The decreasingness of γ implies that x i ≤ x i(n) as requested. Then we deduce that if S(a) is covered by the family {x ∈ S(a) : x i ≤ u x}, i ∈ I, then it is covered by the family {x ∈ S(a) : x i(n) ≤ u x}, n = 1, 2, .. too. Notice that the sequence {x i(n) } n≥1 is increasing with respect to the binary relation " ≤ ". In view of (iv), it has an upper bound, say b ∈ S(a) such that x i(n) ≤ b for every n ≥ 1. But then b does not belong to {x ∈ S(a) : x i(n) ≤ u x}, n = 1, 2, ... Consequently S(a) is P-complete and by Theorem 8, the set S(a) has a maximal point which is also a maximal point of X.

  then it is a rest point. 4) If the variable preference is weak consistent and if an equivalent class [x * ] is u -maximal on [X], then x * (and any element of this class) is ex-ante maximal on X. Proof. Equivalence between (a) and (b) follows from the definition. Equivalence between (b) and (c) follows from the relation (3). The proof of the second assertion uses the same argument. For the third assertion we assume that x * is a ≤ u -rest point. If for some y ∈ X one has x * ≤ x * y, then x * ≤ u y. By the hypothesis x * = y. Hence x * is a rest point of X. To prove the last assertion, assume that the class [x * ] is u -maximal. Suppose that x * ≤ x * y for some y ∈ X. Then [x * ] u [y]. By u -maximality, we have [y] u [x * ], which implies y ≤ u x * . Hence there is an upward path from y to x

* :

Appendix

Proof of Theorem 8. The "only if " part is evident, by taking a among maximal points. To prove the "if" part we first consider the case where the upper transitive closure " ≤ u " is a partial order on X. As we have seen in Proposition 4 this is the case when the variable preference is acyclic. We wish to prove that the set X has ex-ante maximal points. Suppose to the contrary that it is not the case. Then the set P of all strictly increasing nets in S(a) is nonempty. Introduce a partial order on P by inclusion, i.e. for a, b ∈ P one writes a b if and only if a ⊆ b as sets. It is easy to see that this order is a partial order on P and one may apply Zorn's lemma to obtain a maximal element of P. We give here a detailed proof of application of Zorn's lemma. First we prove that P satisfies the hypothesis of Zorn's lemma : every chain D = {a λ : λ ∈ Λ} ⊆ P has an upper bound. Indeed, denote by B the family of all finite subsets of Λ. For each B ∈ B we set a B := λ∈B a λ . It is evident that a B ∈ P. Now we put a 0 = ∪{a B : B ∈ B}. We show that a 0 is a strictly