A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficien...

Given continuous functions M and N of two variables, it is shown that if in a continuous iteration semigroup with only (M,N)-convex or (M,N)-concave elements there are two (M,N)-affine elements, then M = N and every element of the semigroup is M -affine. Moreover, all functions in the semigroup either are M -convex or M -concave.

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q is simplicial, and a finitely g...

We apply theorems of Fischer, Morris and Shapiro on affine semigroup rings to show that if a certain affine semigroup ring defined by a poset is a complete intersection, then the poset is either unicyclic or contains a chain, the removal of which increases the number of connected components of the Hasse diagram. This is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner [2]. We...

The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of A(Bn), the affine nearsemiring over a Brandt semigroup Bn. It is ascertained that both the semigroup reducts of A(Bn) are syntactic semigroups.

— In these notes we study combinatorial and algebraic properties of affine semigroups and their algebras: (1) the existence of unimodular Hilbert triangulations and covers for normal affine semigroups, (2) the Cohen–Macaulay property and number of generators of divisorial ideals over normal semigroup algebras, and (3) graded automorphisms, retractions and homomorphisms of polytopal semigroup al...

An analogue of the Kunz-Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings. Let S be a commutative semigroup (we always assume that S contains a neutral element), and K a field. For every m 6 Z+ the assignment x H-» x, x £ S, induces a K-endomorphism 7m of the semigroup ring R = K[S]. Therefore we can consid...