A Characterization of Maximal Subsemigroups of the Injective Transformation Semigroups with equal Gap and Defect: Boorapa Singha - Team, Support (Thai Journal of Mathematics, 2022)
Suppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. It is known that the semigroup A(X) consisting of all αI(X) such that |Xα|= |Xanα| is a factorisable inverse subsemigroup of I(X). In 2009, all maximal subsemigroups of A(X) has been described when X is uncountable. So, it is an obvious question to ask what happens when X is countably infinite. In this paper, we answer this question by classifying all prime ideals of A(X) and apply these results to characterize all maximal subsemigroups of A(X) for an arbitrary infinite set X. Our results generalize and simplify the results obtained in 2009.