Maximal inverse subsemigroups of \(S(S^ n)\) (Q1061871)

For X a topological space, S(X) will denote the semigroup, under composition, of all continuous functions from X into X. In this paper we concentrate on maximal inverse subsemigroups of \(S(S^ n)\) where \(S^ n\) denotes the n-dimensional sphere. We concentrate particularly on ones with a finite number of idempotents. For instance, we characterize all maximal inverse subsemigroups of S(X) which have exactly two idempotents. We define several types of maximal inverse subsemigroups which have a smallest idempotent. Several open problems are mentioned.