The concept of signed poset has recently been introduced by V. Reiner as a generalization of that of ordinary poset (partially ordered set). We consider the problem of finding a minimum-weight ideal of a signed poset. We show a representation theorem that there exists a bijection between the set of all the ideals of a signed poset and the set of all the "reduced ideals" (defined here) of the as...

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n...

A subset S of a poset (partially ordered set) is convex if and only if S contains every poset element which is between any two elements in S. Poset convex subsets arise in applications that involve precedence constraints, such as in project scheduling, production planning, and assembly line balancing. We give a strongly polynomial time algorithm which, given a poset and element weights (of arbi...

Poset game, which includes some famous games. e.g., Nim and Chomp as sub-games, is an important two-player impartial combinatorial game. The rule of the game is as follows: For a given poset (partial ordered set), each player intern chooses an element and the selected element and it’s descendants (elements succeeding it) are all removed from the poset. A player who choose the last element is th...

Hierarchical data, such as Partially Ordered Set (POSET) is tremendously used in relational databases, especially in data mining and data warehouse based-applications. Unfortunately, SQL (Structured Query Language) does not effectively support hierarchical data structure to manage this sort of data, for example, in Oracle, a CONNECT BY operator is used to query data organized into trees, howeve...

Let SN(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 SN(SN(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 P0 := P and consisting at step m of adding a...

In the realm of linking networks to real world, connectivity (strength connectedness) plays a crucial role. this article, we introduce three types vertices based on indegree and outdegree directed rough fuzzy (DRFNs). We also define concept strength-reducing sets ( $$\mathcal {SRS}s$$ ) DRF-vertices, DRF-edges, important related results using strongest path in graphs (DRFGs). generalize idea Me...

Let V be an n-dimensional vector space over a finite field Fq and P = {1, 2, . . . , n} a poset. We consider on V the poset-metric dP . In this paper, we give a complete description of groups of linear isometries of the metric space (V, dP ), for any poset-metric dP . We show that a linear isometry induces an automorphism of order in poset P , and consequently we show the existence of a pair of...

We generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in...

We study properties of a poset generating a Cohen-Macaulay algebra with straightening laws (ASL for short). We show that if a poset P generates a Cohen-Macaulay ASL, then P is pure and, if P is moreover Buchsbaum, then P is Cohen-Macaulay. Some results concerning a Rees algebra of an ASL defined by a straightening closed ideal are also established. And it is shown that if P is a Cohen-Macaulay ...