Melanija Mitrović: An introduction to a development of the theory of constructive semigroups with apartness

Abstract:

The purpose of this lecture is an introduction to a constructive development of the theory of semigroups with apartness. The focus is on E. Bishop's approach to constructive mathematics (BISH), [1]. The theory of constructive semigroups with apartness is a new approach to semigroup theory and not a new class of semigroups.

Content of lecture:

Part 1. Semigroups - Classical background: Our work is, of course, partly inspired by classical semigroup theory. Algebraic theory of semigroups is even within CLASS a relative new discipline, with the theory proper developing in the second half of the twentieth century. Classically, a semigroup is a set with an associative binary operation defined on it. Introductional and basic study of semigroups includes a description of special elements, special subsets, ideals, Green's relations, homomorphisms, congruences and their relationships. This part of the talk will be mostly based on [6].

Part 2. Semigroups – Intuitionistic Setting: Constructive semigroups with apartness distinguishes from their classical ’friends’ by two significant aspects: first, we use intuitionistic logic rather than classical, secondly, our work is based on the notion of apartness (between elements, elements and sets). In the context of semigroup with apartness we examine the basic notions of special subsets and special relations. Constructive analogues of some classical theorems like, for example, Isomorphism theorems and Cayley's theorem, will be given too. This part of the talk will be based on material given in [3,4]. Important source of ideas, notions and notations of our work is [2]. An example of application(s) of these ideas can be found in [5].

References:

- E. Bishop: Foundations of Constructive Analysis, McGraw-Hill, New York (1967).

- D. S. Bridges, L. S. Vita: Apartness and Uniformity - A Constructive Development, CiE series on Theory and Applications of Computability, Springer, 2011.

- S. Crvenković, M. Mitrović, D. A. Romano: Semigroups with Apartness, Mathematical Logic Quarterly, 59 (6), 2013, 407-414.

- S. Crvenković, M. Mitrović, D. A. Romano: Basic Notions of (Constructive) Semigroups with Apartness, Semigroup Forum, June 2016, Volume 92, Issue 3, 659-674.

- H. Geuvers, R. Pollack, F. Wiedijk, J. Zwanenburg: A Constructive Algebraic Hierarchy in Coq, J. Symbolic Computation (2002) 34, 271-286.

- M. Mitrović: Semilattices of Archimedean Semigroups, University of Niš - Faculty of Mechanical Engineering, Niš (2003).