In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if only the associated partition are Cohen-Macaulay. Both notions of being Cohen–Macaulay admit different refinements: our goal here to link two these refinements. We more precisely prove any (basic-set) whose isomorphism-compatible CL-shellings admits Poincaré–Birkhoff–Witt basis. Furthermore,...

We prove a conjecture of Thomas Lam that the face posets stratified spaces planar resistor networks are shellable. These called uncrossing partial orders. This shellability result combines with Lam's previous these same Eulerian to imply they CW posets, namely regular complexes. Certain subposets orders shown be isomorphic type A Bruhat order intervals; our shelling is coincide on intervals whi...

In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets of nontrivial elements in G. We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply conn...

Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres (“the sphere minus a facet collapses to a (d − 2)-complex”) and for d-balls. In...

In this paper, we give a sufficient condition for a set G of polynomials to be a Gröbner basis with respect to a given term-order for the ideal I that it generates. Our criterion depends on the linkage pattern of the ideal I and of the ideal generated by the initial terms of the elements of G. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider ...

Written by prominent experts in the field, this monograph provides the first comprehensive and unified presentation of the structural, algorithmic, and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions, especially disjunctive and conjunctive normal form representations. It presentswithin this framework the fundamental elements...