We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding...

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J ≡ (J, ⪯). One of themainmotivations for the study is an application ofmatrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spec...

Abstract The explicite formulas for Möbius function and some other important elements of the incidence algebra of an arbitrary cobweb poset are delivered. For that to do one uses Kwaśniewski’s construction of his cobweb posets [8, 9]. The digraph representation of these cobweb posets constitutes a newly discovered class of orderable DAG’s [12, 6, 1] named here down KoDAGs with a kind of univers...

In this work we construct global resolutions for general coherent equivariant sheaves over toric varieties. For this, we use the framework of sheaves over posets. We develop a notion of gluing of posets and of sheaves over posets, which we apply to construct global resolutions for equivariant sheaves. Our constructions give a natural correspondence between resolutions for reflexive equivariant ...

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Möbius function and k-Eulerian posets, which are 2k-thick. Several characterizations of...

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x) − hL(y)| ≤ k, where hL(x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the posets of linear discrepancy 2. We show t...

In this paper, we have established bi-approximation semantics for lattice-based logics with the De Morgan negation (unbounded orthologic), and their morphisms. In addition, we have discussed the dual representation between unbounded ortholattices with strict homomorphisms and polarity frames and d-morphisms. Apart from the abstract construction of dual algebras in the series of the present auth...

Conditions are given on a lattice polytope P of dimension m or its associated affine semigroup ring which imply inequalities for the h∗-vector (h∗0, h∗1, . . . , h∗m) of P of the form hi ≥ hd−i for 1 ≤ i ≤ bd/2c and hbd/2c ≥ hbd/2c+1 ≥ · · · ≥ hd, where hi = 0 for d < i ≤ m. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included.

We give pseudo-LYM inequalities in some posets and give a new restriction in this way for their antichains. Typically these posets fail the LYM inequality and some of them are known not to be Sperner.