The monoid consisting of Kuratowski operations

TL;DR

This paper thoroughly analyzes the algebraic structure of Kuratowski operations, providing a complete Cayley table and classifying semigroups, with some results specific to topology.

Contribution

It constructs the Cayley table for Kuratowski operations and classifies all semigroups and isomorphic types using purely elementary techniques.

Findings

Complete Cayley table for Kuratowski operations

Classification of semigroups and isomorphic types

Identification of topology-specific properties

Abstract

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphic types are applied. Some results apply only to topology, one can not bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.