# Axiomatization of if-then-else over monoids of possibly non-halting programs and tests

## TL;DR

This paper extends the algebraic framework for modeling if-then-else constructs over non-halting programs by introducing $C$-monoids, providing a canonical representation for certain classes.

## Contribution

It generalizes $C$-sets to $C$-monoids, incorporating program composition, and offers a canonical representation for $C$-monoids with adas $C$-algebras.

## Key findings

- Canonical representation of $C$-monoids with adas $C$-algebras
- Extension of $C$-sets to include program composition
- Framework for axiomatizing if-then-else over non-halting programs

## Abstract

In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, the notion of $C$-sets was introduced in the literature by considering the tests from an abstract $C$-algebra. This paper extends the notion of $C$-sets to $C$-monoids which include the composition of programs as well as composition of programs with tests. For the class of $C$-monoids where the $C$-algebras are adas a canonical representation in terms of functional $C$-monoids is obtained.