Automorphism-primal algebras generate verbose varieties

A finite algebra is called automorphism-primal if its clone of term operations coincides with all operations that preserve its automorphisms. We prove that the variety generated by an automorphism-primal algebra is verbose, that is, on every member algebra, every fully invariant congruence is verbal. In [2] we discussed the notions of verbal and fully invariant congruences and examined the relationship between these two concepts. In particular, we called an algebra verbose if every fully invariant congruence is verbal. A variety is verbose if every member algebra has that property. In the earlier paper, we set ourselves the task of understanding verbose varieties. Among the results in [2], we proved that every subalgebra-primal algebra generates a verbose variety. This suggests the conjecture that every quasiprimal algebra generates a verbose variety. While we are still unable to verify that conjecture in complete generality, we are able to prove another special case, namely that every automorphism-primal algebra generates a verbose variety. The purpose of this note is to provide that proof. This result was announced at the conference on General Algebra and Its Applications in Melbourne, 2013. The proof utilizes NU-duality as developed by Brian Davey and his collaborators. We happily dedicate this paper to Brian on the occasion of his birthday and retirement. We begin by formulating the two concepts that are our primary interest here. These notions are presented in isolation in this paper. For a more thorough explanation, consult [2]. See [1] for any concepts or results not fully described in this paper. Definition 1. Let A be an algebra and V a variety of the same similarity type as A. • ΛV = { θ ∈ Con(A) : A/θ ∈ V }. • λV = ⋂ ΛV . • A congruence relation of the form λV for some variety, V, is called a verbal congruence. Definition 2. Let A be an algebra. Write End(A) for the set of endomorphisms of A. A congruence relation, θ, on A is called fully invariant if and only if ∀s ∈ End(A) (x, y) ∈ θ =⇒ ( s(x), s(y) ) ∈ θ. 2 Clifford Bergman Algebra univers. It is well-known that every verbal congruence is fully invariant. The converse is false in general. That phenomenon forms the root of our inquiry. We call an algebra verbose if every fully invariant congruence is verbal. A variety is verbose if every member algebra is verbose. A finite algebra M is called automorphism-primal if its clone of term operations coincides with the clone of all operations that preserve Aut(M). A classical theorem due to Werner [4, 1.14(5)], asserts that a finite algebra M is automorphism-primal if and only if it is quasiprimal, every subuniverse is the set of fixed points of a group of automorphisms, and every isomorphism between nontrivial subalgebras extends to an automorphism of M. An isomorphism between two subalgebras of an algebra M is called an internal isomorphism. We denote by Iso(M) the set of internal isomorphisms of M, viewed as unary partial operations on M , and by Aut(M) the set of automorphisms of M, viewed as unary total operations. As an example, any finite field is automorphism-primal. In this case, the discriminator term is q(x, y, z) = (x − y)n−1 · (x − z) + z, where n is the cardinality of the field. The only subuniverses are the subfields and the only internal isomorphisms are the Frobenius automorphisms, which extend to an automorphism on any field extension. As we remarked above, every automorphism-primal algebra is quasiprimal. We remind the reader that quasiprimal algebras have very special properties. They generate varieties that are semisimple and congruence-distributive, in fact, they have a majority term. Consequently, there are only finitely many subdirectly irreducible algebras, and they are all finite, simple subalgebras of M. Finally, the variety generated by a quasiprimal algebra, M, coincides with SP(M), see [1, Sec. 6.1]. Quasiprimal algebras are strongly dualizable [3, Theorem 3.3.13], however, we need only a “plain vanilla” duality, which is easier to state and will presumably generalize more readily. For the remainder, let M denote an automorphism-primal algebra and V the variety generated by M. Lemma 3. Let M be automorphism-primal and M = 〈M,Aut(M), T 〉 in which T is the discrete topology. Then M yields a duality on V. Proof. By [3, Theorem 3.3.12], every subalgebra of M is either an internal isomorphism or of the form Q1 × Q2, for subalgebras Q1,Q2 of M. The NU-duality theorem [3, Theorem 2.3.4] tells us that V is dualized by 〈M,Sub(M), T 〉. Thus we need only use automorphism-primality to show that Aut(M) entails Iso(M) and Q1 ×Q2, for every Q1, Q2 ≤M. All of the entailments are easily obtained from [3, Theorem 2.4.5]. For every automorphism g, fix(g) is entailed by g. By Werner’s theorem, every subalgebra of M is of the form ⋂ i fix(gi), so they are all entailed. Therefore, every product Q1×Q2 is entailed. Finally, suppose h : Q1 → Q2 is an internal isomorphism. If Q1 is nontrivial, then by Werner result cited above, h = g Q1 Vol. 00, XX Automorphism-primal algebras generate verbose varieties 3 for some g ∈ Aut(M), so h is entailed. On the other hand, if Q1 is trivial, then h = Q1 ×Q2, so, as we have already shown, h is entailed. We need to review a few details of the duality guaranteed by Lemma 3. Let X denote the category of closed subspaces of powers of M. There is a pair of functors