We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.2010 Mathematics Subject Classification. 05E40, 13C14, 13D02.

Abstract. We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of powers of monomial ideals in terms of their linear relation graph, and apply this to show that the index of depth stability and the index of stability for the associated prime ideals of polymatroidal ideals is bounded by their analytic spread.

Abstract. In this paper we show that polyomino ideal of a simple polyomino coincides with the toric ideal of a weakly chordal bipartite graph and hence it has a quadratic Gröbner basis with respect to a suitable monomial order.

Abstract. We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a squarefree Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced.

Let L be a distributive lattice and R(L) the associated Hibi ring. We compute reg R(L) when L is a planar lattice and give bounds for reg R(L) when L is non-planar, in terms of the combinatorial data of L. As a consequence, we characterize the distributive lattices L for which the associated Hibi ring has a linear resolution.1991 Mathematics Subject Classification. 05E40,13D02,16E05.