Topological and ditopological unosemigroup (Q2443484)

The authors introduce a new topological algebraic structure called ditopological unosemigroup. This is a topological semigroup endowed with continuous unary operations of (left, right) units. Introducing topological unosemigroups was motivated by the problem of generalization of Hryniv's Embedding Theorems beyond the class of compact topological inverse semigroups. Let \(S\) be a semigroup. A unary operation \(\lambda:S\rightarrow S\) is called left unit operation on \(S\) if \(\lambda(x)x=x\) for all \(x\in S\). A semigroup \(S\) endowed with a left unit operation \(\lambda:S\rightarrow S\) is called a left unosemigroup. A topological left unosemigroup is a topological semigroup \(S\) endowed with a continuous left unit operation \(\lambda:S\rightarrow S\). A topological right unosemigroup is defined in a similar way. A topological unosemigroup is a topological semigroup \(S\) endowed with a continuous left unit operation \(\lambda:S\rightarrow S\) and a continuous right unit operation \(\rho:S\rightarrow S\). In this paper, ditopological unosemigroups, uniformizable topological unosemigroups, ditopological inverse semigroups and operations over ditopological unosemigroups (for example, the semidirect product of topological unosemigroups and the Hartman-Mycielski extension) are introduced.