Constructive semigroups with apartness -- a state of the art

This chapter aims to provide a clear and understandable picture of constructive semigroups with apartness in Bishop's style of constructive mathematics, BISH. Our theory is partly inspired by the classical case, but it is distinguished from it in two significant aspects: we use intuitionistic logic rather than classical throughout; our work is based on the notion of apartness (between elements of the set, and, later, between elements and its subsets). Following Heyting, at least initially, classical semigroup theory is seen as a guide that helps us to develop the constructive theory of semigroups with apartness. To have a structure, we need a set, a relation, and rules establishing how we will put them together. Working within classical or intuitionistic logic, in order to analyze algebraic structures, it is necessary to start with study on sets and ordered sets, relational systems, etc. A comparative analysis between presented classical and constructive results is also a part of this chapter. All proofs can be found in the Appendix.