Abstract. We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if it does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.
In this paper we study sub-semigroups of a finite or an infinite zero-divisor semigroup S determined by properties of the zerodivisor graph (S). We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. In particular, we discover a class of sub-semigroups of reduced semigroups and we study properties of sub-semigroups of finite or infinite semilattices with the least element. As an application, we provide a characterization of the graphs which are zero-divisor graphs of Boolean rings. We also study how local property of (S) affects global property of the semigroup S, and we discover some interesting applications. In particular, we find that no finite or infinite two-star graph has a corresponding nil semigroup.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.