Topic
Cancellative semigroup
About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.
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6 citations
TL;DR: In this article, the maximal regular subsemigroups of when is a finite-dimensional subspace of over a finite field were derived and the rank and idempotent rank for when is an -dimensional subspaces of an -deterministic vector space over a fixed field.
Abstract: Let be a vector space and let denote the semigroup (under composition) of all linear transformations from into . For a fixed subspace of , let be the semigroup consisting of all linear transformations from into . In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc. 77(3) (2008), 441–453] proved that is the largest regular subsemigroup of and characterized Green’s relations on . In this paper, we determine all the maximal regular subsemigroups of when is a finite-dimensional subspace of over a finite field. Moreover, we compute the rank and idempotent rank of when is an -dimensional subspace of an -dimensional vector space over a finite field .
6 citations
TL;DR: In this paper, the smallest digital semigroup containing a set of positive integers was computed and a connection between the digital semigroups and LD-semigroups was established between them.
6 citations
TL;DR: In this article, the fundamental double four-spiral semigroup (DSP4) was introduced as a building block of bisimple, idempotent-generated regular semigroups.
Abstract: We give a new description of the fundamental double four-spiral semigroup. The fundamental four-spiral semigroup Sp4 and the fundamental double fourspiral semigroup DSp4 were introduced in [1], [3], and [4]. These semigroups are interesting examples of fundamental regular semigroups, and are indispensable building blocks of bisimple, idempotent-generated regular semigroups. Their basic properties are recalled in parts 1 and 2 of this note. In part 3 we give an alternate construction ofDSp4 in terms of the free semigroup on two generators, as a set of quadruples with a simple, bicyclic-like multiplication. This permits shorter proofs and easier access to the main properties of DSp4 :d escriptions of DSp4=L and DSp4=R (part 4); reduced form of the elements (part 5); and the property of congruencesC6Lthat DSp4=C is completely simple (part 6).
6 citations
01 Dec 1970
TL;DR: In this paper, the authors consider a semigroup whose algebraic structure may be isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group.
Abstract: If we consider a semigroup, its algebraic structure may be such that it is isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group. This problem was investigated in 1931 by Ore who obtained in (4) a set of necessary conditions for this embedding. A necessary condition is that the semigroup should be cancellative: for any a, x, y in the semigroup either xa = ya or ax = ay implies that x = y. Malcev in (3) showed that this was not sufficient. It is enough to note that his example was a non-commutative semigroup: a commutative cancellative semigroup is embeddable algebraically in a group.
6 citations