Topological and ditopological unosemigroups

TL;DR

This paper introduces the concept of (di)topological unosemigroups, a new topological algebraic structure with continuous unary operations, and explores their properties and closure under various algebraic operations.

Contribution

It defines and studies (di)topological unosemigroups, expanding the understanding of topological semigroups with continuous unit operations and their algebraic and topological properties.

Findings

Includes all topological groups and semilattices

Closed under subunosemigroup, product, and extension operations

Contains all compact and uniformizable topological unosemigroups

Abstract

In this paper we introduce and study a new topologo-algebraic structure called a (di)topological unosemigroup. This is a topological semigroup endowed with continuous unary operations of left and right units (which have certain continuous division property called the dicontinuity). We show that the class of ditopological unosemigroups contains all topological groups, all topological semilattices, all uniformizable topological unoid-semigroups, all compact topological unosemigroups, and is closed under the operations of taking subunosemigroup, Tychonoff product, reduced product, semidirect product, and the Hartman-Mycielski extension.