The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

Abstract

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor.The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid P, if toric ideal of P is generated by double swaps then toric ideal of any expansion of P has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.

On the Facet Ideal of an Expanded Simplicial Complex

“…A special case of their result is the second part of Corollary 3.5. Then in [13], the authors studied the expansion of monomial ideals in the concept of Bayati and Herzog. They showed that a monomial ideal has linear quotients if and only if its expansion does (c.f.…”

“…They showed that a monomial ideal has linear quotients if and only if its expansion does (c.f. [13,Theorem 1.7]). Also, a monomial ideal is weakly polymatroidal if and only if its expansion is (c.f.…”

“…Also, a monomial ideal is weakly polymatroidal if and only if its expansion is (c.f. [13,Theorem 1.4]). As a consequence of these results we have:…”

Expansion and contraction functors on matriods

“…In this regard studying the structural properties of matroids from different point views have been considered by many researchers. Also classifying matroids with a desired property or making modifications to a matroid so that it satisfies a special property is the subject of many research papers, see for example [1,7,9,11,15,17,20].…”

“…The notion of expansion is a known notion appeared in different terminologies as parallelization or duplication in combinatorics [5,4,13,16]. In [15], the authors studied behaviors of expansion functor on some algebraic structures associated to discrete polymatroids. It was shown that a nonempty finite set is a discrete polymatroid if and only if its an arbitrary expansion is a discrete polymatroid (c.f.…”