In this paper we study topological properties of the poset of injective words and the lattice of classical non-crossing partitions. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This extends the well-known result that those posets are shellable. Both results rely on a new poset fiber theorem, for doubly homo...

We study signed differential posets, a signed version of Stanley’s differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [3, 4, 6]. We show that these identities result from a signed differen...

First, Cogis and Habib (RAIRO Inform. 7Mor. 13 (1979), 3-18) solved the jump number problem for series-parallel partially ordered sets (posets) by applying the greedy algorithm and then Rival (Proc. Amer. Math. Sot. 89 (1983). 387-394) extended their result to N-free posets. The author (Order 1 (1984), 7-19) provided an interpretation of the latter result in the terms of arc diagrams of posets ...

A poset is (3 + 1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the (3 + 1)-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have enumerated graded (3+1)-free posets, but until now the general enumeration problem has remained open. We enumerate all (3 + 1)-free posets by giving a decomposit...

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to...

In this paper, we first consider (po-)torsion free and principally weakly (po-)flat $S$-posets, specifically  we discuss when (po-)torsion freeness implies principal weak (po-)flatness. Furthermore, we give a counterexample to show that Theorem 3.22 of Shi is incorrect. Thereby we present a correct version of this theorem. Finally, we characterize pomonoids over which all cyclic $S$-posets are ...

In this paper we study the notions of cogenerator and subdirectlyirreducible in the category of S-poset. First we give somenecessary and sufficient conditions for a cogenerator $S$-posets.Then we see that under some conditions, regular injectivityimplies generator and cogenerator. Recalling Birkhoff'sRepresentation Theorem for algebra, we study subdirectlyirreducible S-posets and give this theo...

In this article we describe a very efficient method to construct pairwise nonisomorphic posets (equivalently, T0 topologies). We also give the results obtained by a computer program based on this algorithm, in particular the numbers of nonisomorphic posets on 15 and 16 points and the numbers of labelled posets and topologies on 17 and 18 points.

We study a class of algebras associated with linear spaces and its relations with polymatroids and integral posets, i.e. posets supporting homogeneous ASL. We prove that the base ring of a transversal polymatroid is Koszul and describe a new class of integral posets. As a corollary we obtain that every Veronese subring of a polynomial ring is an ASL.

Two partially ordered monoids S and T are called Morita equivalent if the categories of right S-posets and right T -posets are Pos-equivalent as categories enriched over the category Pos of posets. We give a description of Pos-prodense biposets and prove Morita theorems I, II, and III for partially ordered monoids.