Towards a pseudoequational proof theory

Abstract

A new scheme for proving pseudoidentities from a given set Σ of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when Σ defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when Σ defines a pseudovariety V which is σ-reducible for the equation x = y, provided Σ is enough to prove a basis of identities for the variety of σ-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.

Equational Axiomatization of Algebras with Structure

“…Theorem 4.4 can be used to recover several known sound and complete equational logics, but it also applies to settings where no such logic is known, for instance, a logic of profinite equations (however, cf. recent work of Almeida and Klíma [5]). In each case, the challenge is to translate our two abstract proof rules into concrete syntax, which requires the identification of a syntactic equivalent of the two properties of an equational theory.…”

Profinite congruences and unary algebras

“…Second, a modification of the first example to make it a fully invariant closed congruence that turns out to be a profinite congruence on a finitely generated relatively free profinite semigroup whose quotient is countable and contains infinitely many idempotents. Third, an example, which is again derived from the first example, of a fully invariant closed congruence on a relatively profinite unary algebra which is not profinite, thereby giving a negative answer to the general case of our question raised in [6]. The latter example prompts a study of the relationship between profinite monoids and profinite unary algebras which leads us to exhibit a pair of adjoint functors between the two categories.…”

“…On the other hand, one may describe "constructively" the smallest closed congruence on Ω X V containing R by applying the following natural "procedure" considered in [6]:…”

“…In all the exceptions found so far, the reason why such a description is known is that, unlike what happens in general, closed congruences are profinite. In the realm of semigroup theory, many favorable examples are presented in [6], where the aim is to find a general method to determine when a given pseudoidentity is a consequence, in all finite models, of a given set of pseudoidentities. Such examples are of special type in the sense that fully invariant closed congruences on relatively free profinite semigroups are concerned.…”

A note on the finite basis and finite rank properties for pseudovarieties of semigroups

“…A major difference between dealing with identities and pseudoidentities is the absence of an analog for pseudoidentities of the completeness theorem for equational logic [1, Theorem 3.8.8], in the sense that there is no complete and sound finite deductive system for pseudoidentities. Instead, we invoke the general theory developed in [1, Section 3.8] for which an alternative has recently been proposed in the form of a deductive system involving infinite proofs [2].…”