Semigroups and Their Subsemigroup Lattices pp 105-126 | Cite as

# Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups

## Abstract

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 Subi*S* 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 Subi*S* is a sublattice of Sub*S*. It is easy to see that Sub*S* = Subi*S* 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 Subi*G* 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.

## Keywords

Maximal Subgroup Inverse Semigroup Simple Semigroup Finite Width Subgroup Lattice## Preview

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