On scattered convex geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice $\Omega(\eta)$, that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.Comment: 25 pages, 1 figure, submitte

Length of an intersection

A poset \bfp is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, \ell(\bfp) of \bfp. We prove that if the vertex set $X$ of \bfp is infinite, of cardinality $\kappa$, and the ordering $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting \ell(X,\leq_i)=\kappa\multordby q_i+r_i, with $r_i<\kappa$, denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~: \ell(\bfp)< \kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ where $|\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).Comment: 13 page

N-free extensions of posets.Note on a theorem of P.A.Grillet

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.Comment: 7 pages, 4 picture

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.Comment: 27 pages; submitte