DEFINABILITY IN THE LATTICE OF EQUATIONAL THEORIES OF SEMIGROUPS

We study first-order definability in the lattice L of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable in L. As examples, if T is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T(partial derivative)} is definable, where T(partial derivative) is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable.