Categorical Equivalence of Algebras with a Majority Term

Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection. The main theorem is used to get a different sort of characterization of categorical equivalence for algebras generating an arithmetical variety. We also consider clones of co-height at most two. In addition, we provide new proofs of several characterizations in the literature, including quasi-primal, lattice-primal and congruence-primal algebras. Majority operations have long held a special place in universal algebra. It has been known for quite some time that any variety of algebras possessing a majority term is congruence distributive. In 1975, Baker and Pixley discovered that for a finite algebra A with a majority term, the set of subalgebras of A completely determines the term operations on A. In addition, every subalgebra of A (with k ≥ 2) is completely determined by all of its 2-fold projections. Conversely, G. Bergman proved that, under some obviously necessary consistency conditions, every family of subalgebras of A is obtained from a subalgebra of A by 2-fold projections. By universal algebraic standards, algebras with a majority term are not rare. Any structure possessing a lattice reduct has a majority term, as does any quasiprimal algebra. More generally, any algebra that generates an arithmetical variety (i.e. both congruence distributive and congruence permutable) will have such a term. It is customary to consider term-equivalence (of algebras or varieties) as a fundamental relationship in universal algebra. Indeed, term-equivalent varieties are usually treated as interchangeable. From this perspective, the Baker-Pixley result mentioned above tells us that if A is a finite algebra with a majority term, then V(A) is completely determined by the set Sub(A). However, it is also natural to consider a variety as a category of algebras, in which the arrows are exactly the homomorphisms. With this as our starting point, the central relationship between varieties becomes equivalence of categories, which 1991 Mathematics Subject Classification. Primary 08C05, 08B10; Secondary 08A02, 08A40, 18C05.