Universal equational theories and varieties of algebras

Equational Logic and Equational Theories of Algebras

Universal Unification and Regular Equational ACFM Theories

Lattices of Equational Theories as Church Algebras

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 u...

Characterization of Protomodular Varieties of Universal Algebras

Protomodular categories were introduced by the first author more than ten years ago. We show that a variety V of universal algebras is protomodular if and only if it has 0-ary terms e1, . . . , en, binary terms t1, . . . , tn, and (n+1)-ary term t satisfying the identities t(x, t1(x, y), . . . , tn(x, y)) = y and ti(x, x) = ei for each i = 1, . . . , n.

Decision Problems for Equational Theories of Relation Algebras

The foundation of an algebraic theory of binary relations was laid by C. S. Peirce, building on earlier work of Boole and De Morgan. The basic universe of discourse of this theory is a collection of binary relations over some set, and the basic operations on these relations are those of forming unions, complements, relative products (i.e., compositions), and converses (i.e., inverses). There is...