Composition of Terms and Essential Positions in Deduction

The paper deals with inductive, positional and Σ−compositions of terms in order to describe some varieties defined by such compositions. In algebra, equational logic, theoretical computer science, etc. the composition of terms is defined as replacements of some variables by external terms. We consider the positional composition of terms as a tool, which allows us to extend the derivation rules in formal deduction of identities. Cases when closure operators of the sets of identities of a given type are fully invariant congruences are investigated. 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. TSR-closure of the sets of identities is obtained by applying inductive and positional term composition in deduction of identities. 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 Σ of identities.