arXiv subjects
Publications and source records attributed to Maurice Pouzet.
We are interested in overgroups of the automorphism group of the Rado graph. One class of such overgroups is completely understood; this is the class of reducts. In this article we tie recent work on various other natural overgroups, in particular establishing group connections between them and the reducts.
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 of a wonderful Hausdorff? type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010]. Key words and phrases. posets, lattices, convex sublattice, retracts, fixed point property
We consider the transformation reversing all arcs of a subset $X$ of the vertex set of a tournament $T$. The \emph{index} of $T$, denoted by $i(T)$, is the smallest number of subsets that must be reversed to make $T$ acyclic. It turns out that critical tournaments and $(-1)$-critical tournaments can be defined in terms of inversions (at most two for the former, at most four for the latter). We interpret $i(T)$ as the minimum distance of $T$ to the transitive tournaments on the same vertex set, and we interpret the distance between two tournaments $T$ and $T'$ as the \emph{Boolean dimension} of a graph, namely the Boolean sum of $T$ and $T'$. On $n$ vertices, the maximum distance is at most $n-1$, whereas $i(n)$, the maximum of $i(T)$ over the tournaments on $n$ vertices, satisfies $\frac {n-1}{2} - \log_{2}n \leq i(n) \leq n-3$, for $n \geq 4$. Let $ \mathcal{I}_{m}^{< ω}$ (resp. $\mathcal{I}_{m}^{\leq ω}$) be the class of finite (resp. at most countable) tournaments $T$ such that $i(T) \leq m$. The class $\mathcal {I}_{m}^{< ω}$ is determined by finitely many obstructions. We give a morphological description of the members of $\mathcal {I}_{1}^{< ω}$ and a description of the critical obstructions. We give an explicit description of an universal tournament of the class $\mathcal{I}_{m}^{\leq ω}$.
This paper is a contribution to the study of a quasi-order on the set $Ω$ of Boolean functions, the \emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\tildeΩ$. 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 $\tildeΩ$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\tildeΩ$.
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-semilattice of compact elements of $L$, is well-founded and contains neither $[ω]^{<ω}$, nor $\underlineΩ(ω^*)$ as a join-subsemilattice. 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 $I_{<ω}(Q)$, 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.
We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements. We show that for every chain $α$ of size $κ$, there is a set $\B$ of at most $2^κ$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(α)$ if and only if the join-semilattice $K(L)$ of its compact elements contains no join-subsemilattice isomorphic to a member of $\B$. We show that among the join-subsemilattices of $[ω]^{<ω}$ belonging to $\B$, one is embeddable in all the others. We conjecture that if $α$ is countable, there is a finite set $\B$.
We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain.
We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation.
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
A tournament is \emph{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 {\it 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 $ϕ_T(n)\simeq an^k$, 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 \emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\tildeΩ$. 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 $\tildeΩ$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\tildeΩ$.
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 $\mathcal C$ 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 $m\geq 3$ if and only if there is a Boolean group $G$ on $A$ for which $\mathcal C$ is the set of all operations $f(x_1,..., x_n)$ of the form $a+\sum_{i\in I}x_i$ for $a\in A$ and $I\subseteq \{1,..., n\}$.
P.J.Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure $R$ is an integral domain if and only if $R$ is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.
The {\it profile} of a relational structure $R$ is the function $ϕ_R$ which counts for every integer $n$ the number of its $n$-element substructures up to an isomorphism. Many counting functions are profiles. Interesting examples come from permutation groups. Some salient facts about the behavior of the profile are presented. Techniques from ordered sets and combinatorics (notably the notion of well-quasi-order, the related notions of ordered algebras, Ramsey theorem) are illustrated. Ongoing resarch suggests to view the profile of a relational structure $R$ as the Hilbert function of some graded algebra associated with $R$. A hint at the solution of a conjecture of P.J.Cameron on the integrity of the ring of the orbit algebra is given. Recent progress made with Y.Boudabbous and N.Thiéry on the conjecture that the profile is a quasi-polynomial if its growth is polynomial (and the structure has a finite kernel) are presented.
A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric space embeds isometrically into an indivisible ultrametric metric space if and only if it does not contain a strictly increasing sequence of balls.
Posets which are retract of products of chains are characterized by means of two properties: \emph{the chain-gap property} and \emph{the selection property} (Rival and Wille, 1981 \cite {R-W}). Examples of posets with the selection property and not the chain-gap property are easy to find. To date, the Boolean lattice $\Power (ω_1)/Fin$ was the sole example of lattice without the selection property \cite {R-W}. We prove that it does not have the chain-gap property. We provide an example of a lattice which has the chain-gap property but not the selection property. This answer questions raised in \cite {R-W}.
The profile of a relational structure R is the function phi_R which counts for every integer n the number, possibly infinite, phi_R(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. Several graded algebras can be associated with R in such a way that the profile of R is simply the Hilbert function. An example of such graded algebra is the age algebra introduced by P.~J.~Cameron. In this paper, we give a closer look at this association, particularly when the relational structure R decomposes into finitely many monomorphic components. In this case, several well-studied graded commutative algebras (e.g. the invariant ring of a finite permutation group, the ring of quasi-symmetric polynomials) are isomorphic to some age algebras. Also, phi_R is a quasi-polynomial, this supporting the conjecture that, with mild assumptions on R, phi_R is a quasi-polynomial when it is bounded by some polynomial.
It was proved few years ago that classes of Boolean functions definable by means of functional equations \cite{EFHH}, or equivalently, by means of relational constraints \cite{Pi2}, coincide with initial segments of the quasi-ordered set $(Ω, \leq)$ made of the set $Ω$ of Boolean functions, suitably quasi-ordered. The resulting ordered set $(Ω/\equiv, \sqsubseteq)$ embeds into $([ω]^{<ω}, \subseteq)$, the set -ordered by inclusion- of finite subsets of the set $ω$ of integers. We prove that $(Ω/\equiv, \sqsubseteq)$ also embeds $([ω]^{<ω}, \subseteq)$. We prove that initial segments of $(Ω, \leq)$ which are definable by finitely many obstructions coincide with classes defined by finitely many equations. This gives, in particular, that the classes of Boolean functions with a bounded number of essential variables are finitely definable. As an example, we provide a concrete characterization of the subclasses made of linear functions.