Soft Bitopological Groups via Soft Elements

TL;DR

This paper introduces soft bitopological groups, combining soft set theory with bitopological groups, providing a unified framework that simplifies continuity proofs and explores fundamental properties and examples.

Contribution

It develops the theory of soft bitopological groups, characterizing their structure, continuity, and separation axioms, and provides illustrative examples of their properties.

Findings

Soft bitopological groups are characterized by continuity of specific maps.

Translations and inverses are homeomorphisms in each induced topology.

The paper establishes separation axioms and compactness results for these groups.

Abstract

Working in the soft-element (classical) viewpoint, we introduce \emph{soft bitopological groups}: soft groups endowed with two soft topologies such that the induced topologies on the set of soft elements make the soft-element group into a (classical) bitopological group. This approach unifies and simplifies continuity proofs, because the group operations become coordinatewise and standard topological-group methods apply. We organize the theory in a standard ``definitions--characterizations--properties--examples'' format. In particular, we (i) record the induced topology and induced bitopology on soft elements of a soft set; (ii) characterize soft bitopological groups by continuity of the map $(a,b)\mapsto a\ast b^{-1}$ in each induced topology; (iii) show that translations and inversion are homeomorphisms in each induced topology; (iv) collect pairwise soft separation axioms and pairwise soft compactness results (including the finiteness principle for compactness when the parameter set is finite); and (v) define soft bitopological group homomorphisms and basic invariants. Several examples illustrat that the two topologies can be independent (non-comparable) even in Hausdorff situations.