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EXTENSIONS OF CLIFFORD SUBSEMIGROUPS OF THE FINITE SYMMETRIC INVERSE SEMIGROUP

ABSTRACT We describe maximal Clifford subsemigroups of the finite symmetric inverse semigroup I n of the set X n − { 1, 2,…,n } and obtain their complete classification, determine the number and the order of such subsemigroups, give a criterion for two maximal Clifford subsemigroups of I n to be isomorphic, and determine (up to isomorphism) the number of such subsemigroups. Further, both maximal ideal extensions and maximal nil extensions of any Clifford subsemigroup S of I n are described. In particular, we show that S has a unique maximal ideal extension in I n and, up to isomorphism, S has a unique maximal nil extension in I n .