Equality-Test and If-Then-Else Algebras: Axiomatization and Specification

Abstract

An equality-test algebra has a two-element Boolean sort and an equality-test operation $eq_s $ for each non-Boolean sort s, where $eq_s (x,y)$ equals TRUE if $x = y$ and FALSE otherwise. An if-then-else algebra is an equality-test algebra with the if-then-else operations $[\_,\_,\_]_s $ adjoined: $[b,x,y]_s $ equals x if $b = \text{TRUE}$ and y if $b = \text{FALSE}$. A finite set of axioms for the conditional-equational (i.e., quasi-equational) theory of equality-test algebras is given. A finite axiomatization of the equational theory of if-then-else algebras is also given, and it is shown that this also serves as a basis for the conditional-equational theory of if-then-else algebras. Finite bases for the equational theories of several classes of algebras closely related to if-then-else algebras were previously known. The power of conditional and equational specifications of equality-test and if-then-else data types are investigated, and the following results, among others, are obtained. (i) Every equalit...