Maurice Pouzet - Academia.edu (original) (raw)

Papers by Maurice Pouzet

Cornell University - arXiv, Dec 5, 2021

This paper is a contribution to the study of hereditary classes of relational structures, these c... more This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ruškuc and Vatter. In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fraïssé. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized. We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of N ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs.

A poset is well-partially ordered (WPO) if all its linear extensions are well orders ; the suprem... more A poset is well-partially ordered (WPO) if all its linear extensions are well orders ; the supremum of ordered types of these linear extensions is the length, ℓ() of . We prove that if the vertex set X of is infinite, of cardinality κ, and the ordering ≤ is the intersection of finitely many partial orderings ≤_i on X, 1≤ i≤ n, then, letting ℓ(X,≤_i)=κ q_i+r_i, with r_i<κ, denote the euclidian division by κ (seen as an initial ordinal) of the length of the corresponding poset :ℓ()< κ⊗_1≤ i≤ nq_i+ |∑_1≤ i≤ n r_i|^+ where |∑ r_i|^+ denotes the least initial ordinal greater than the ordinal ∑ r_i. This inequality is optimal (for n≥ 2).

Contributions Discret. Math., 2018

We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join... more We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join semi-lattice of compact elements of L, is well founded and contains neither [\omega]^\omega, nor \underscore(\Omega)(\omega*) as a join semilattice. As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is well founded and contains no infinite independent set. If K(L) is a join-subsemilattice of I_{<\omega}(Q), the set of finitely generated initial segments of a well founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered.

Israel Journal of Mathematics, 1978

Due to an unfortunate error, pages 73 and 74 were reversed.

Canadian Journal of Mathematics, 1986

The purpose of this article is to develop aspects of a classification theory for reflexive graphs... more The purpose of this article is to develop aspects of a classification theory for reflexive graphs. A first important step was already taken in [2]; throughout we follow, at least the spirit, of the classification theory for ordered sets initiated in [1]. For a graph G let V(G) denote its vertex set and E(G) ⊆ V(G) × V(G) its edge set. A graph K is a subgraph of G if V(K) ⊆ V(G) and for a, b ∊ V(K), (a, b) ∊ E(K) just if (a, b) ∊ E(G). The subgraph K of G is a retract of G, and we write K ◅ G, if there is an edge-preserving map g of V(G) to V(K) satisfying g(v) = v for each v ∊ V(K); g is called a retraction. A reflexive graph is an undirected graph with a loop at every vertex. The reason for a loop at a vertex is that an edge-preserving map can send the two vertices of an adjacent pair to it. The concept is illustrated in Figure 1. From here on, though, we shall for convenience suppress the illustration of the loops in the figures of reflexive graphs.

Annals of Pure and Applied Logic, 2001

Abstract The age of a relational structure R is the set A (R) of finite restrictions of R conside... more Abstract The age of a relational structure R is the set A (R) of finite restrictions of R considered up to isomorphism. R. Fraisse, who introduced this notion, showed that ages coincide with nonempty ideals of the poset consisting of finite relational structures, considered up to isomorphism and ordered by embeddability. Here, given two ages A ⊆ B , we study the poset D ( A , B ) consisting of ages C in sandwich between A and B . Among other things we show that if D ( A , B ) is infinite then it contains an infinite chain.

Annales de l’institut Fourier, 1972

Algebra universalis, 2010

In 1986, the second author classified the minimal clones on a finite universe into five types. We... more In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a "small" clone of one of the five types. From it we deduce, in part, an earlier result, namely that if C is a clone on a universe A with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some m-ary idempotent operation is not a projection some m ≥ 3 if and only if there is a Boolean group G on A for which C is the set of all operations f (x 1 ,. .. , x n) of the form a + i∈I x i for a ∈ A and I ⊆ {1,. .. , n}.

The workshop ”Homogeneous Structures, A Workshop in Honour of Norbert Sauer‘s 70th Birthday” took... more The workshop ”Homogeneous Structures, A Workshop in Honour of Norbert Sauer‘s 70th Birthday” took place at the Banff International Research Station from November 8th to November 13th, 2015. The purpose of the following note is to gather the list of open problems that were posed during the problem sessions. Contributions appear in alphabetical order, according to the name of their autho

A system RRR of equivalence relations on a set AAA (with at least 333 elements) is \\emph{semirig... more A system RRR of equivalence relations on a set AAA (with at least 333 elements) is \\emph{semirigid} if only the trivial operations (that is the projections and constant functions) preserve all members of RRR. To a system RRR of equivalence relations we associate a graph GRG_RGR. We observe that if RRR is semirigid then the graph GRG_RGR is 222-connected. We show that the converse holds if all the members of RRR are atoms of the lattice EE E of equivalence relations on AAA. We present a notion of graphical composition of semirigid systems and show that it preserves semirigidity.

We characterize pairs of orthogonal countable ordinals. Two ordinals α and β are orthogonal if th... more We characterize pairs of orthogonal countable ordinals. Two ordinals α and β are orthogonal if there are two linear orders A and B on the same set V with order types α and β respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if α and β are two countable ordinals, with α≤β, then α and β are orthogonal if and only if either ω + 1≤α or α =ω and β < ωβ.

A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than... more A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The profile of a tournament T is the function ϕ_T which counts for each integer n the number ϕ_T(n) of tournaments induced by T on the n-element subsets of T, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of ϕ_T is either polynomial, in which case ϕ_T(n)≃ an^k, for some positive real a, some non-negative integer k, or as fast as some exponential.

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the... more This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset Ω̃. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of Ω̃ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of Ω̃.

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a ... more The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application ...

Let S_N(P) be the poset obtained by adding a dummy vertex on each diagonal edge of the N's of... more Let S_N(P) be the poset obtained by adding a dummy vertex on each diagonal edge of the N's of a finite poset P. We show that S_N(S_N(P)) is N-free. It follows that this poset is the smallest N-free barycentric subdivision of the diagram of P, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with P_0:=P and consisting at step m of adding a dummy vertex on a diagonal edge of some N in P_m, proving that the result of this algorithm does not depend upon the particular choice of the diagonal edge choosen at each step. These results are linked to drawing of posets.

In 2014, Cégielski, Grigorieff and Guessarian characterized unary self-maps on the set Z of integ... more In 2014, Cégielski, Grigorieff and Guessarian characterized unary self-maps on the set Z of integers which preserve all congruences of the additive group. In this note, we propose a shorter and straigthforward proof. We replace this result in the frame of universal algebra.

A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on resul... more A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.

We investigate infinite versions of vector and affine space partition results, and thus obtain ex... more We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible.

We extend to binary relational systems the notion of compact and normal structure, introduced by ... more We extend to binary relational systems the notion of compact and normal structure, introduced by J.P.Penot for metric spaces, and we prove that for the involutive and reflexive ones, every commuting family of relational homomorphisms has a common fixed point. The proof is based upon the clever argument that J.B.Baillon discovered in order to show that a similar conclusion holds for bounded hyperconvex metric spaces and then refined by the first author to metric spaces with a compact and normal structure. Since the non-expansive mappings are relational homomorphisms, our result includes those of T.C.Lim, J.B.Baillon and the first author. We show that it extends the Tarski's fixed point theorem to graphs which are retracts of reflexive oriented zigzags of bounded length. Doing so, we illustrate the fact that the consideration of binary relational systems or of generalized metric spaces are equivalent.

A sibling of a relational structure R is any structure S which can be embedded into R and, vice v... more A sibling of a relational structure R is any structure S which can be embedded into R and, vice versa, in which R can be embedded. Let sib(R) be the number of siblings of R, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, sib(R) is either 1, countably infinite, or the size of the continuum; but even showing the special case sib(R)=1 or infinite is unsettled when R is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree T the number of trees which are sibling of T is either one or infinite. We prove that if R is countable and _0-categorical, then indeed sib(R) is one or infinite. Furthermore, sib(R) is one if and only if R is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thiéry 2013 and studied...