Semigroups ofI-Type

Journal of Algebra, 1998

Bulletin of the American Mathematical Society, 1970

Lecture Notes in Computer Science, 2004

The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8-element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator.

Algebra Universalis, 1988

Advances in Mathematics, 1973

Semilattice-ordered semigroup is an important algebraic structure. It is ordered semigroup under anti-order. Some basic properties of semillatice-ordered semigroups with apartness are given by constructive point of view. Let I and K be compatible an ideal and an anti-ideal of semilattice-ordered semigroup S. Constructions of compatible congruence E(I) and anti-congruence Q(K) on S, generated by I and K respectively, are given. Besides, we construct compatible order T and anti-order θ T on factor-semigroup S/(E(I), Q(K)). Some basic properties of such constructed semigroups are given.

Semigroup Forum, 2013

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid.

Archive for history of exact sciences, 2009

In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues.

Proceedings of the Japan Academy, 1972

Eprint Arxiv Math 0210217, 2002

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Etingof, Schedler and Soloviev, Gateva-Ivanova and Van den Bergh, and the authors considered special classes of such algebras in the contexts of noetherian algebras, Grobner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation.