Products of two idempotent transformations over arbitrary sets and vector spaces

Abstract

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

Products of three idempotent transformations

“…For finite X, the authors of [6] characterised when a € T(X) can be written as a product of two or of three idempotents in T(X), and later Saito [9] did the same for products of four idempotents in T(X). In [10] Thomas extended some of the earlier work to arbitrary sets and proved a corresponding result for products of two idempotent linear transformations of a vector space. In Section 2, we characterise products of three idempotents in T(X) when X is arbitrary.…”

“…For convenience we state Thomas' characterisation of products of two idempotents in T(X) [10,Theorem 2]. We say an idempotent S € T{X) is proper if S ^ idx (recall that no injective transformation of X can be a product of proper idempotents).…”