On the Cartesian product of the Menger algebras of terms and relational formulas

Abstract

A relational formula which is a first-order formula that only uses relation symbols and terms of arbitrary type is one of the important concepts in the study of algebras and algebraic systems. In this paper, necessary and sufficient conditions for any element in a semigroup whose universe arises from the Cartesian product of the Menger algebras of terms and relational formulas to be idempotent and 2-potent are given. By the formula for counting the occurrence of all variables in a formula \(F\), we further show that the order of such pairs is 1, 2, or infinite.