Soft Topological Modules via Soft Elements

Subhasis Ray

doi:10.55195/jscai.1896971

Soft Topological Modules via Soft Elements

Abstract

Working in the soft-element (classical) viewpoint, we introduce and study \emph{soft topological modules}. Given a soft ring $S$ and a soft $S$-module $F$ on a parameter set $A$, the set of soft elements $\SE(F)$ is naturally a module over the ring of soft elements $\SE(S)$ under pointwise operations. We define an induced ``element topology'' on $\SE(F)$ as the box topology generated by coordinatewise soft open sets. Our main characterization theorem shows that a pair $(F,\tau)$ is a soft topological $S$-module if and only if the induced module $\big(\SE(F),\tau^\ast\big)$ is a (classical) topological module over $\big(\SE(S),\rho^\ast\big)$. This approach reduces most proofs to standard arguments in topological algebra. We further develop basic properties of soft topological modules, including translation invariance, a continuity criterion using the subtraction map, behavior of submodules and quotients, functoriality of $\SE$, and transfer of separation and compactness properties (with a finiteness principle when the parameter set is finite). Examples are included to illustrate the theory

Keywords

Supporting Institution

Visva-Bharati University, Santiniketan, India (institutional support only; no external funding).

Ethical Statement

This manuscript is a purely theoretical study in soft set theory/topology and does not involve human participants, animals, or any personal/sensitive data. Therefore, ethical approval and informed consent are not applicable. The author declares no conflict of interest.

Thanks

The author thanks Visva-Bharati University for providing research facilities.

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Details

Primary Language

English

Subjects

Fuzzy Computation, Soft Computing

Journal Section

Research Article

Authors

Subhasis Ray ^*
0000-0003-1100-4632
India

Publication Date

June 6, 2026

Submission Date

February 24, 2026

Acceptance Date

April 23, 2026

Published in Issue

Year 2026 Volume: 7 Number: 1

DOI

https://doi.org/10.55195/jscai.1896971

IZ

https://izlik.org/JA84XW83NW

Cite

RIS / Bibtex

APA

Ray, S. (2026). Soft Topological Modules via Soft Elements. Journal of Soft Computing and Artificial Intelligence, 7(1), 14-24. https://doi.org/10.55195/jscai.1896971

AMA

1.Ray S. Soft Topological Modules via Soft Elements. JSCAI. 2026;7(1):14-24. doi:10.55195/jscai.1896971

Chicago

Ray, Subhasis. 2026. “Soft Topological Modules via Soft Elements”. Journal of Soft Computing and Artificial Intelligence 7 (1): 14-24. https://doi.org/10.55195/jscai.1896971.

EndNote

Ray S (June 1, 2026) Soft Topological Modules via Soft Elements. Journal of Soft Computing and Artificial Intelligence 7 1 14–24.

IEEE

[1]S. Ray, “Soft Topological Modules via Soft Elements”, JSCAI, vol. 7, no. 1, pp. 14–24, June 2026, doi: 10.55195/jscai.1896971.

ISNAD

Ray, Subhasis. “Soft Topological Modules via Soft Elements”. Journal of Soft Computing and Artificial Intelligence 7/1 (June 1, 2026): 14-24. https://doi.org/10.55195/jscai.1896971.

JAMA

1.Ray S. Soft Topological Modules via Soft Elements. JSCAI. 2026;7:14–24.

MLA

Ray, Subhasis. “Soft Topological Modules via Soft Elements”. Journal of Soft Computing and Artificial Intelligence, vol. 7, no. 1, June 2026, pp. 14-24, doi:10.55195/jscai.1896971.

Vancouver

1.Subhasis Ray. Soft Topological Modules via Soft Elements. JSCAI. 2026 Jun. 1;7(1):14-2. doi:10.55195/jscai.1896971