Rank properties of the semigroup of endomorphisms over Brandt semigroup

Abstract

Since the work of Marczewski [10], many authors have studied the rank properties in the context of general algebras (cf. [1–3,9,11,12,14]). The concept of rank for general algebras is analogous to the concept of dimension in linear algebra. The dimension of a vector space is the maximum cardinality of an independent subset, or equivalently, it is the minimum cardinality of a generating set of the vector space. A subset U of a semigroup is said to be independent if every element ofU is not in the subsemigroup generated by the remaining elements of U, i.e., ∀a ∈ U, a /∈ U \ {a} . It can be observed that the minimum size of a generating set need not be equal to the maximum size of an independent set in a semigroup. Accordingly, Howie and Ribeiro have considered various concepts of ranks for a finite semigroup (cf. [5,6]). 1. r1( ) = max{k: every subset U of cardinality k in is independent}. 2. r2( ) = min{|U| : U ⊆ , U = }. 3. r3( ) = max{|U| : U ⊆ , U = ,U is independent}. 4. r4( ) = max{|U| : U ⊆ ,U is independent}.