Theory of constructive semigroups with apartness -- foundations, development and practice

TL;DR

This paper reviews and advances the theory of constructive semigroups with apartness, addressing open problems and exploring applications across mathematics, computer science, and social sciences in Bishop's constructive framework.

Contribution

It provides solutions to open problems in constructive semigroup theory and clarifies its foundations and applications in various disciplines.

Findings

Solutions to open problems in constructive semigroup theory

Potential applications in multiple scientific fields

A clear presentation in Bishop's constructive style

Abstract

This paper has several purposes. We present through a critical review the results from already published papers on the constructive semigroup theory, and contribute to its further development by giving solutions to open problems. We also draw attention to its possible applications in other (constructive) mathematics disciplines, in computer science, social sciences, economics, etc. Another important goal of this paper is to provide a clear, understandable picture of constructive semigroups with apartness in Bishop's style both to (classical) algebraists and the ones who apply algebraic knowledge.