Bicyclic semigroup

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

Semigroups generated by a group and an idempotent

Abstract: It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n − 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n − 1 while the analogous result is true for the semigroup of all n × n matrices over a field. In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ϵ . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow. In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ϵ is an idempotent of rank n − 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup. The fourth section deals with a special ca...

Semigroups of nonnegative matrices

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 .

Subordination in the sense of bochner and a related functional calculus

Abstract: We prove a new representation of the generator of a subordinate semigroup as limit of bounded operators. Our construction yields, in particular, a characterization of the domain of the generator. The generator of a subordinate semigroup can be viewed as a function of the generator of the original semigroup. For a large class these functions we show that operations at the level of functions has its counterpart at the level of operators.