Essential variables and positions in terms

TL;DR

This paper introduces the concept of $\Sigma$-composition of terms and explores essential variables and positions to simplify formal deduction of identities, including the notion of $\Sigma$-balanced identities.

Contribution

It defines $\Sigma$-composition and essential positions, and introduces $\Sigma$-balanced identities, advancing the formal deduction process in algebraic identities.

Findings

Defined $\Sigma$-composition as replacement between $\Sigma$-equal terms.

Introduced and analyzed $\Sigma$-balanced identities.

Showed how these concepts simplify formal deduction of identities.

Abstract

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