Bicyclic Work done on a Star-like Finite Chain Semigroup.

This paper compares the work of bicyclic products and idempotent product chains on the star-like operator to demonstrate potential connections between semigroup and group theory. This research advances our understanding of star-like transformation semigroups and their algebraic features, particularly in terms of bicyclic and idempotent products. The findings add to previous understanding by obtaining the arithmetic and polynomial sequential patterns. We show that the composition of any star-like transformations produces the bicyclic products work done such that , for all . The research contributes to the larger discipline of abstract algebra by introducing new results and approaches for examining algebraic structures.