Lattice isomorphisms of modular inverse semigroups

Abstract

For an inverse semigroup S we will consider the lattice of inverse subsemigroups of S, denoted L(S). A major problem in algebra has been that of finding to what extent an algebra is determined by its lattice of subalgebras. (See, for example, the survey article [9]). By a lattice isomorphism (L-isomorphism, structural isomorphism, or projectivity) of an inverse semigroup S onto another T we shall mean an isomorphism Φ of L(S) onto L(T). A mapping φ from S to T is said to induce Φ if AΦ = Aφ for all A in L(S). We say that S is strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T.