Inverse Semigroups

Abstract

Recall that for inverse semigroups S and T an isomorphism of the lattice SubiS on the lattice SubiT 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.