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Showing papers on "Bicyclic semigroup published in 1979"


Journal ArticleDOI
TL;DR: In this paper, the authors consider a semigroup satisfying the property abc = ac and prove that it is left semi-normal and right quasi-normal, where ac is the number of variables in the semigroup.
Abstract: This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

246 citations


Book ChapterDOI
01 Jan 1979
TL;DR: In this article, the authors discuss the algebraic theory of semigroups of nonnegative matrices, and the most important applications of the material in the chapter involve the solvability of certain nonnegative matrix equations arising in the areas of mathematical economics and mathematical programming.
Abstract: This chapter discusses the semigroups of nonnegative matrices. The matrix multiplication is associative, and the product of two non-negative matrices is again a nonnegative matrix. The most important applications of the material in the chapter involve the solvability of certain nonnegative matrix equations arising in the areas of mathematical economics and mathematical programming. A subgroup G of a semigroup T is called a maximal subgroup of T if it is not properly contained in any other subgroup of T . This chapter discusses the algebraic theory of semigroups of nonnegative matrices. Idempotent elements play a fundamental role in algebraic semigroup theory. Every matrix in the semigroup N n of nonnegative matrices is not regular. The tool to be used to characterize Green's relations for regular elements in N n would be that of rank. However, this tool is more of a vector space notion and is too sophiscated to characterize Green's relations on the entire semigroup N n .

75 citations


Journal ArticleDOI

40 citations


Journal ArticleDOI
TL;DR: The semigroup of all operators T such that ( Tx, Tx )⩾( x, x ), for all elements of x of a finite-dimensional complex vector space with (, ) a given, possibly indefinite Hermitian form on that space, is the object under study in this article.

23 citations



Journal ArticleDOI
TL;DR: The notions of synchronization and simplification with respect to a given subsemigroup P of a semigroup S in terms of the syntactic semigroup of P are described to give a unified account of several theorems previously published.

15 citations


Journal ArticleDOI
TL;DR: In this paper, a Galois connection between the set of relations on the social relations themselves and the lattice of congruences on the universal semigroup is applied to the evolution of kinship systems.

10 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that an E-m semigroup is a disjoint union of power-joined semigroups, and that strongly reversible semigroup connections with power-joints are also possible.
Abstract: In this note we study some properties of E-m semigroups, recently defined by Nordahl in [3]. In particular we study connections with power joined and strongly reversible semigroups. We prove, among other things, that an E-m semigroup is a disjoint union of power joined semigroups.

8 citations



Journal ArticleDOI

6 citations


Journal ArticleDOI
TL;DR: In this article, the quadratic case was studied, i.e., the case when the polynomials are of degree at most two, and the resulting semigroup was called a quadrastic semigroup.


Journal ArticleDOI
TL;DR: In this paper, a ring of quotients of the semigroup ring R(S) is discussed, where R has a σ-set Σ and S has a ΃set Δ, and where R is an integral domain and S is a commutative cancellative semigroup.
Abstract: A ring of quotients of the semigroup ring R(S) is discussed where R has a σ-set Σ and S has a σ-set Δ. In particular, we study the cases where (1) R is an integral domain and S is a commutative cancellative semigroup, (2) R is a commutative ring and S is a semilattice and (3) R is a commutative ring and S is a Rees matrix semigroup over a semigroup.