Characterization of binary polynomials of idempotent algebras

Abstract

In this paper, we characterize the set of all binary algebraic (or polynomial) operations of an idempotent algebra that has at least one r-ary algebraic operation, (r ≥ 2), depending on every variable such that there is no an (r + 2)-ary algebraic operation depending on at least (r + 1) variables. We prove that this set forms a finite Boolean algebra, and then we characterize this Boolean algebra.