Morita Equivalence of Almost-primal Clones

Two algebraic structures A and B are called categorically equivalent if there is a functor from the variety generated by A to the variety generated by B, carrying A to B, that is an equivalence of the varieties when viewed as categories. We characterize those algebras categorically equivalent to A when A is an algebra whose set of term operations is as large as possible subject to constraints placed on it by the subalgebra or congruence lattice of A, or the automorphism group of A. Two categories C and D are said to be equivalent if there are functors F : C → D and G : D → C such that the composite functors F ◦ G and G ◦ F are naturally isomorphic to the identities on D and C respectively. It is natural to ask whether some property of an object, morphism or an entire category is preserved under every equivalence of categories. Moreover, given an object (or morphism, or category), one might wish to characterize the class of objects obtained by applying all equivalences to that starting object. Any variety of algebras (that is, a class of algebras closed under the formation of subalgebra, product, and homomorphic image) forms a category, in which the morphisms are taken to be all homomorphisms between algebras. A surprising number of “algebraic” properties have been shown to be preserved under equivalence, when the domain of categories is restricted to varieties of algebras. Some of these are familiar to anyone who has worked with categories of algebras, such as Cartesian products and homomorphism kernels; others are somewhat unexpected. Examples of the latter are ‘surjective homomorphism’ and ‘finite algebra’. A large collection of examples of this phenomenon can be found in [5]. A classical example of categorical equivalences of varieties of algebras is Morita’s Theorem, which provides necessary and sufficient algebraic conditions on two rings with unit in order for their varieties of unitary modules to be equivalent as categories. Other instances of categorical equivalence of varieties have been discovered using the tools of duality theory as in [5] and [17]. There are also examples of objects characterized up to categorical equivalence in the literature. One of the most striking is a result of Hu’s [14]. A finite, nontrivial algebra A is called primal if every operation on the universe of A is a term operation of A. For example, the two-element Boolean algebra is primal. Hu’s theorem for primal algebras asserts that if P is a primal algebra, then the class of all algebras of the form F (P) as F ranges through all equivalences between varieties, is This paper was written while the first author was visiting the University of Illinois at Chicago.