# Portugaliae Mathematica

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**Volume 75, Issue 2, 2018, pp. 79–119**

**DOI: 10.4171/PM/2012**

Published online: 2018-12-12

Towards a pseudoequational proof theory

Jorge Almeida^{[1]}and Ondřej Klíma

^{[2]}(1) Universidade do Porto, Portugal

(2) Masaryk University, Brno, Czechia

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

*Keywords: *Pseudoidentity, syntactical proof, semigroup, profinite monoid, completeness, reducible pseudovariety, implicit signature

Almeida Jorge, Klíma Ondřej: Towards a pseudoequational proof theory. *Port. Math.* 75 (2018), 79-119. doi: 10.4171/PM/2012