Perfect Product of two Squares in Finite Full Transformation Semigroup

Abstract

In this paper, we investigate the concept of the perfect product of two squares in the context
of finite full transformation semigroups. We provide a comprehensive analysis of the conditions
under which the product of two idempotent elements in a transformation semigroup forms a
perfect product of two squares. Specifically, we examine the relationship between the kernel
and image of idempotents, as well as the interplay between the domain and image of these
transformations. The main result establishes that for two idempotent elements α and β in Tn,
if the domain and image of α and β satisfy certain equivalence conditions, then their product is
a perfect product of two squares. We also explore related properties of disjoint cycles and how
these contribute to the structural characteristics of the semigroup. Our findings extend the
existing theory of transformation semigroups and offer valuable insights into the decomposition
of semigroup elements into squares, contributing to the broader field of semigroup theory.