Essential variables and positions in terms

The paper deals with Σ−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. 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.