Let S-Set be the category of $S$-sets, sets together with the actions of a semigroup $S$ on them. And, let S-Pos be the category of $S$-posets, posets together with the actions compatible with the orders on them. In this paper we show that the category S-Pos is a radical extension of S-Set; that is there is a radical on the category S-Pos, the order desolator radical, whose torsion-free class i...

This paper is the third in a series of manuscripts that examine combinatorics Kunz polyhedron $P_m$, whose positive integer points are bijection with numerical semigroups (cofinite subsemigroups $\mathbb Z_{\ge 0}$) smallest element $m$. The faces $P_m$ indexed by family finite posets (called posets) obtained from divisibility lying on given face. In this paper, we characterize to what extent m...

We show that for every orthomodular poset P=(P,≤,′,0,1) of finite height there can be defined two operators forming an adjoint pair with respect to order-like relation on the power set P. This enables us introduce so-called operator residuated corresponding P from which original recovered. this construction applied also weakly and dually posets. Examples such posets are included.

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Faà di Bruno Hopf algebra.

We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable cofinality remains open.

Counting the linear extensions of a partially ordered set (poset) is a fundamental problem with several applications. We present two exact algorithms that target sparse posets in particular. The first algorithm breaks the counting task into subproblems recursively. The second algorithm uses variable elimination via inclusion–exclusion and runs in polynomial time for posets with a cover graph of...

We provide a combinatorial recipe for constructing all posets of height at most two which the corresponding type-A Lie poset algebra is contact. In case that such are connected, discrete Morse theory argument establishes posets' simplicial realizations contractible. It follows from cohomological result Coll and Gerstenhaber on semi-direct products contact algebras absolutely rigid.

We study a birational map associated to any finite poset P . This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P . Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams...

The Kahn--Saks inequality is a classical result on the number of linear extensions finite posets. We give new proof this for posets width two using explicit injections lattice paths. As consequence we obtain $q$-analogue, multivariate generalization and an equality condition in case. also discuss conditions general prove several implications between conjectured to be equivalent.

Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank ...