Soft Bitopological Groups via Soft Elements
S. Ray
TL;DR
This work introduces soft bitopological groups via the soft-element viewpoint, proving that such a structure is equivalent to a classical bitopological group on the set of soft elements $\mathrm{SE}(F)$ equipped with two induced topologies $(\tau_1)^*$ and $(\tau_2)^*$. This perspective reduces many proofs to standard topological-group arguments, yielding translations and inversion as homeomorphisms and characterizations through the continuity of $\Delta_i(a,b)=a\ast b^{-1}$. The authors develop pairwise soft separation axioms, pairwise soft compactness, and connectedness, with componentwise criteria ensuring the properties hold on each $F(t)$. They also define soft bitopological group homomorphisms and examine invariants, compactness, and connectedness under morphisms, illustrating the theory with several examples, including cases of incomparable induced topologies even in Hausdorff settings. The paper closes with potential extensions to soft Haar measures, quotients and completions, and generalizations to fuzzy/rough/neutrosophic frameworks, highlighting the practical impact of transferring soft-structure problems to classical bitopological settings.
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.