Cancellative semigroup

About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.

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

Right commutative semigroups

Abstract: In this chapter we deal with semigroups which satisfy the identity axy = ayx. These semigroups are called right commutative semigroups. It is clear that a right commutative semigroup is medial and so we can use the results of the previous chapter for right commutative semigroups. For example, every right commutative semigroup is a semilattice of right commutative archimedean semigroups and is a band of right commutative t-archimedean semigroups. A semigroup is right commutative and simple if and only if it is a left abelian group. Moreover, a semigroup is right commutative and archimedean containing at least one idempotent element if and only if it is a retract extension of a left abelian group by a right commutative nil semigroup. We characterize the right commutative left cancellative and the right commutative right cancellative semigroups, respectively. Clearly, a semigroup is right commutative and left cancellative if and only if it is a commutative cancellative semigroup. A semigroup is right commutative and right cancellative if and only if it is embeddable into a left abelian group if and only if it is a left zero semigroup of commutative cancellative semigroups. It is shown that a right commutative semigroup is embeddable into a semigroup which is a union of groups if and only if it is right separative.

• primary ideals in quasi-commutative ternary semigeoups

Abstract: I n this paper we study the structure of cancellative quasi-commutative primary ternary semigroups. In fact we prove that if T is a cancellative quasi-commutative ternary semigroup, then (1) S is a primary ternary semigroup (2) proper prime ideals in T are maximal and (3) semiprimary ideals in T are primary, are equivalent.

Matrix representations of ample semigroups

Abstract: An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.