We introduce pretty k-clean monomial ideals and k-decomposable multicomplexes, respectively, as the extensions of the notions of k-clean monomial ideals and k-decomposable simplicial complexes. We show that a multicomplex Γ is k-decomposable if and only if its associated monomial ideal I(Γ) is pretty k-clean. Also, we prove that an arbitrary monomial ideal I is pretty k-clean if and only if its polarization I p is k-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.

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.

In this paper we show that expansion of a Buchsbaum simplicial complex is CMt, for an optimal integer t ≥ 1. Also, by imposing extra assumptions on a CMt simplicial complex, we prove that it can be obtained from a Buchsbaum complex.

We prove that a monomial ideal I generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.2010 Mathematics Subject Classification. 13F20; 05E40.