A Characterization of Maximal Subsemigroups of the Injective Transformation Semigroups with equal Gap and Defect

Boorapa Singha

Abstract

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 $\alpha\in I(X)$ such that $|X\backslash\dom\alpha|= |X\backslash \ran\alpha|$ 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.