Abstract

The paper deals with Σ-composition of terms, which allows us to extend the derivation rules in formal deduction of identities. The concepts of essential variables and essential positions of terms with respect to a set of identities form a key step in the simplification of the process of formal deduction. Σ-composition of terms is defined as replacement between Σ-equal terms. This composition induces ΣR-deductively closed sets of identities. In analogy to balanced identities we introduce and investigate Σ-balanced identities for a given set of identities Σ.

Highlights

  • Let F be any finite set, the elements of which are called operation symbols

  • Let X be a finite set of variables, and let τ be a type with the set of operation symbols F = ∪j≥0Fj

  • An algebra A = A; F A of type τ is a pair consisting of a set A and an indexed set F A of operations, defined on A

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Summary

Introduction

Let F be any finite set, the elements of which are called operation symbols. Let τ : F → N be a mapping into the non-negative integers; for f ∈ F , the number τ (f ) will denote the arity of the operation symbol f. An identity t ≈ s ∈ Id(τ ) is satisfied in the algebra A, if the term operations tA and sA, induced by the terms t and s on the algebra A are equal, i.e., tA = sA In this case we write A |= t ≈ s and if Σ is a set of identities of type τ , A |= Σ means that A |= t ≈ s for all t ≈ s ∈ Σ. We use the concept of essential variables, as defined in [5] and we consider such variables with respect to a given set of identities, which is a fully invariant congruence.

Compositions of terms
Stable varieties and globally invariant congruences

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