Lattices of equational theories as Church algebras
Giulio Manzonetto, Antonino Salibra · 2009
Abstract. We introduce the class of Church algebras, which is general enough to compass all Boolean algebras, Heyting algebras and rings with unit. Using a new equational characterization of central elements, we prove that Church algebras satisfy a Stone representation theorem. We show that every lattice of equational theories is isomorphic to the congruence lattice of a suitable Church algebra, and we use this property to prove a meta-Stone representation theorem which is applicable to all varieties of algebras. We say Σ is an equational theory iff Σ is a set of identities closed under the rules of the equational calculus. The set L(Σ) = {T: Σ ⊆ T, T is an equational theory} forms a lattice under inclusion. L is a lattice of equational theories (etlattice, for short) iff L is isomorphic to the lattice L(Σ) for some equational theory Σ (or dually isomorphic to the lattice of all subvarieties of some variety of algebras). In 1966 A.I. Malcev [3] posed the following question: which lattices can be represented as et-lattices?