On correct structures and similarity measures of soft sets along with historic comments of Prof. D. A. Molodtsov

TL;DR

This paper reviews and clarifies the correct structures and similarity measures of soft sets, addressing past misconceptions and building on Molodtsov's foundational work to improve the theoretical framework of soft set theory.

Contribution

It revisits and corrects the structures and operations of soft sets, providing formal accounts, matrix representations, and similarity measures based on Molodtsov's foundational concepts.

Findings

Corrected the structures and operations of soft sets.

Developed matrix representations for soft sets.

Proposed measures to quantify similarity between soft sets.

Abstract

After the paper of Molodtsov ( D. Molodtsov, Soft set theory first results. Computers and Mathematics with Applications, 37(4-5) (1999), 19-31), soft set theory grew at a breakneck pace. Several authors have introduced various operations, relations, results, as well as other aspects in soft set theory and hybrid structures incorrectly, despite their widespread use in mathematics and allied areas. In his paper (D. A. Molodtsov, Equivalence and correct operations for soft sets. International Robotics and Automation Journal, 4(1) (2018), 18-21), Molodtsov, the father of soft set theory, pointed out the wrong results and notions. Molodtsov (D. A. Molodtsov, Structure of soft sets. Nechetkie Sistemy i Myagkie, 12(1) (2017), 5-18) also stated that the concept of soft set had not been fully understood and used everywhere. As a result, it seemed reasonable to revisit the quirks of those conceptions and provide a formal account of the notion of soft set equivalency. Molodtsov already explored the correct operations on soft sets. We use some notions and results of Molodtsov (Molodtsov D.A. (2017) Structure of soft sets. Nechetkie Sistemy i Myagkie, 12(1) (2017), 5-18) to create matrix representations as well as certain associated operations of soft sets, and to quantify the similarity between two soft sets.