Semigroups and Their Subsemigroup Lattices pp 326-352 | Cite as

# Inverse Semigroups

## Abstract

Recall that for inverse semigroups *S* and *T* an isomorphism of the lattice Subi*S* on the lattice Subi*T* is called a *projectivity* of *S* upon *T*. As we have already mentioned in Subsection 30.3, regarding projectivities of inverse semigroups, we deal with the same basic problems as for the “usual” lattice isomorphisms. We only name them; the reader is referred to the corresponding paragraphs of Subsection 30.2. For a fixed class A of inverse semigroups, the following problems can be stated: the *problem of projective classification*, the *problem of projective determinability*, the *problem of strict projective determinability*, the *problem of inducing mappings*. Observe that here, when speaking about strict projective determinability, there is no need to include in the definition the mention of anti-isomorphisms (compare with 30.1). Indeed, in view of the identity (*xy*)^{−1} = *y* ^{−1} *x* ^{−1}, which holds in inverse semigroups, the mapping *x* ↦ *x* ^{−1} of an arbitrary inverse semigroup is an anti-automorphism; therefore, if *φ*: *S* → *T* is an anti-isomorphism of inverse semigroups, then the mapping s ↦ *φ*(*s*)^{−1} is an isomorphism of *S* onto *T* which induces the same projectivity of *S* upon *T* as *φ* does.

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