Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups
As we mentioned in the Preface, one can treat inverse semigroups as unary semi-groups, i.e. as algebraic systems with two operations: the binary operation of multiplication and the unary operation of taking the inverse element. From this point of view it is natural to associate with any inverse semigroup S the lattice of all subsystems, i.e. unary subsemigroups of S. But a unary subsemigroup of an inverse semigroup is none other than an inverse subsemigroup. So we shall consider the lattice SubiS of all inverse subsemigroups of S (of course, again the empty set is treated as an inverse subsemigroup). In view of the identity (xy)−1 = y −1 x −1, which is valid in any inverse semigroup, the join of two inverse subsemigroups of an inverse semigroup S is an inverse subsemigroup; therefore SubiS is a sublattice of SubS. It is easy to see that SubS = SubiS if and only if S is a periodic Clifford semigroup. This observation will be used several times below without explicit reference. For an arbitrary group G, the lattice SubiG coincides with the lattice of all subgroups of G augmented by the adjoined zero. It follows that, in any general considerations of lattices of inverse subsemigroups, the case when we deal with the subgroup lattice of a group is inevitably involved. As in the case of the usual subsemigroup lattices, in examinations of lattices of all inverse subsemigroups one succeeds not infrequently in realizing a reduction to groups.
KeywordsMaximal Subgroup Inverse Semigroup Simple Semigroup Finite Width Subgroup Lattice
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