Maximal and Minimal Congruences on the Semigroup 𝑇𝐸 (𝑋)

Abstract

In semigroup theory, transformations play a crucial role. This paper explores a specific type of transformation semigroup, denoted by . Here, is a non-empty set, and consists of all transformations on X that preserve the equivalence classes established by an equivalence relation on . We delve into the internal structure of by exploring how to partition its elements into the coarsest and finest possible partitions while preserving the validity of the transformation operation within each partition. These partitions correspond to maximal and minimal congruences on , respectively. We then address the existence of a specific type of congruence on where each equivalence class forms a subsemigroup itself.