Combinatorial results to measure the complexity of quantifier free formulas in algebraic systems

The complexity of terms of type [Formula: see text] of an algebra of type [Formula: see text] can be measured by several methods, including the depth of a term, the total number of variables or operation symbols that appear in a term and the length of a term. Generally, algebraic systems can be viewed as a generalization of algebras. To describe properties of algebraic systems of arbitrary types, quantifier free formulas of type [Formula: see text] induced by terms, relation symbols, and logical connectors are essential. In this paper, we aim to define various measurements of the complexity of quantifier free formulas and then formalize these measurements under different operations. Applications of these explicit formulas for some classes of mappings are provided.