Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.

Congruences on *-Regular Semigroups

Abstract: In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.

The lattice of positive quasi-orders on a semigroup

Abstract: In the present paper we study some properties of positive quasi-orders on simigroups and using these results we describe all semilattice and chain homomorphic images of a semigroup.

Unique prime factorization in a partial semigroup of matrix-polynomials

Abstract: Let K be a field. We consider divisability and factorization into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] and determinant 1, where multiplication is defined as matrix-multiplication if the degrees of the factors add up, cf. Section 2. Our aim is to establish a unique factorization result, cf. Theorem 3.1. Although our considerations are purely algebraic and in fact quite elementary, they should be seen in connection with some results of complex analysis. Let us explain this motivation: Let W (z) = (wij(z))i,j=1,2 be a 2 × 2-matrix function whose entries are entire functions, i.e. are defined and holomorphic in the whole complex plane. We say that W belongs to the class Mκ where κ is a nonnegative integer, if wij(z) = wij(z), W (0) = I, detW (z) = 1, and if the kernel KW (w, z) := W (z)JW (w) − J z − w has κ negative squares. Thereby