Topological and ditopological unosemigroups

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.

💡 Research Summary

The paper introduces a novel topological‑algebraic object called a (di)topological unosemigroup. Formally, a unosemigroup consists of a topological semigroup (S) equipped with two continuous unary operations, a left‑unit map (\lambda:S\to S) and a right‑unit map (\rho:S\to S). These maps satisfy the algebraic identities (\lambda(x)\cdot x = x) and (x\cdot\rho(x)=x) for every (x\in S). In addition to continuity, the authors require a special “dicontinuity” property: for any open set (U\subseteq S), the pre‑images (\lambda^{-1}(U)) and (\rho^{-1}(U)) must be open and must preserve the same topological partition structure that (U) induces. This condition can be viewed as a continuous analogue of a division operation: (\lambda) and (\rho) act as left and right “division by a unit” that respects the ambient topology.

The first major contribution is a systematic identification of existing topological algebraic structures that fit into this framework. Every topological group becomes a ditopological unosemigroup by taking (\lambda(x)=\rho(x)=x^{-1}); the group axioms guarantee both the unit identities and dicontinuity. Topological semilattices (commutative idempotent semigroups) also qualify, with the identity map serving as both (\lambda) and (\rho). Uniformizable topological unoid‑semigroups—structures previously studied in the context of uniform spaces—satisfy the dicontinuity condition because the uniform structure provides a compatible family of neighborhoods that are invariant under the unit maps. Compact topological unosemigroups are automatically ditopological: compactness forces any continuous unary operation to be closed, and the dicontinuity follows from standard compact‑Hausdorff arguments.

Having established a broad class of examples, the authors turn to closure properties. They prove that the class of ditopological unosemigroups is stable under several natural constructions:

1. Sub‑unosemigroups – If (T\subseteq S) is a subsemigroup closed under (\lambda) and (\rho), the restrictions (\lambda|_T) and (\rho|_T) inherit continuity and dicontinuity, making (T) a ditopological unosemigroup.

2. Tychonoff products – For a family ({S_i}_{i\in I}) of ditopological unosemigroups, the product ( \prod_i S_i) equipped with component‑wise multiplication, (\lambda((x_i))=(\lambda_i(x_i))) and (\rho((x_i))=(\rho_i(x_i))) is again a ditopological unosemigroup. The product topology ensures that openness of pre‑images is checked coordinatewise.

3. Reduced products – Given a closed ideal (I) in a ditopological unosemigroup (S), the reduced product (S/I) (identifying all elements of (I) to a single zero) inherits well‑defined unit maps because (\lambda) and (\rho) map (I) into itself. Continuity and dicontinuity descend to the quotient.

4. Semidirect products – If a topological group (G) acts continuously on a ditopological unosemigroup (S) by semigroup automorphisms that commute with (\lambda) and (\rho), the semidirect product (S\rtimes G) carries natural unit maps (\lambda(s,g)=(\lambda(s),g)) and (\rho(s,g)=(\rho(s),g)). The action’s continuity guarantees that the dicontinuity condition is preserved in the combined structure.

5. Hartman‑Mycielski extension – The Hartman‑Mycielski construction (HM(X)) replaces a space (X) by the space of all continuous maps from the unit interval into (X) equipped with the compact‑open topology. When (X) is a ditopological unosemigroup, pointwise multiplication and pointwise application of (\lambda) and (\rho) turn (HM(X)) into a ditopological unosemigroup. The extension is crucial because it provides a method to embed arbitrary topological spaces into contractible ditopological unosemigroups, mirroring the classical use of the Hartman‑Mycielski construction in topological group theory.

The paper’s methodology blends classical topological arguments (e.g., use of compactness, product topology, quotient maps) with algebraic considerations specific to the unit maps. A key technical lemma shows that dicontinuity is equivalent to the requirement that for every open neighbourhood (U) of a point (x), there exist open neighbourhoods (V) of (\lambda(x)) and (W) of (x) such that (V\cdot W\subseteq U). This “local division” property is used repeatedly to verify that the constructions above preserve the ditopological nature.

In the concluding section, the authors discuss potential directions. Since the unit maps behave like continuous division by a left or right identity, they suggest investigating homomorphism theorems, representation theory, and cohomology for ditopological unosemigroups. Moreover, the closure under the Hartman‑Mycielski extension hints at applications to the theory of topological dynamics and to the construction of universal objects in categories of topological semigroups. The paper thus opens a new avenue where algebraic division‑type operations coexist with topological continuity in a robust, categorical framework.