Soft Union-Gamma Product of Groups

Year 2025, Volume: 3 Issue: 2, 58 - 72, 31.12.2025

Zeynep Ay , Aslıhan Sezgin

https://doi.org/10.63063/jsat.1724734

Abstract

Soft set theory constitutes a mathematically rigorous and algebraically expressive framework for modeling systems characterized by uncertainty, vagueness, and parameter-dependent variability. At the heart of this formalism lies a comprehensive repertoire of algebraic operations and binary product constructions that collectively furnish the universe of soft sets with a rich and finely articulated internal algebraic topology. Within this algebraic landscape, we introduce and formally investigate a novel binary operation, termed the soft union–gamma product, defined over soft sets whose parameter domains are endowed with group-theoretic structure. The operation is rigorously constructed within an axiomatic framework that ensures compatibility with generalized notions of soft subsethood and soft equality. A systematic algebraic analysis is undertaken to establish the operation’s fundamental properties, including closure, associativity, commutativity, idempotency, and distributivity over other soft set operations, as well as its behavior in relation to identity and absorbing elements. Moreover, the product’s interaction with both the null and absolute soft sets is rigorously characterized. Two principal contributions emerge from this investigation. First, the integration of the soft union–gamma product enhances the internal operational harmony of soft set theory by embedding it within a formally consistent, axiomatically governed algebraic environment. Second, the operation lays a conceptual and structural foundation for the development of a generalized soft group theory, wherein soft sets indexed by group-structured parameter domains emulate the axiomatic behavior of classical group-theoretic systems through suitably defined soft operations. This study constitutes a substantive advancement in the algebraic consolidation and theoretical generalization of soft set theory.

Keywords

Soft sets , Soft subsets , Soft equalities , Soft union-gamma product

Grupların Esnek Birleşim-Gama Çarpımı

Abstract

Esnek küme teorisi, belirsizlik, muğlaklık ve parametreye bağlı değişkenlik ile karakterize edilen sistemleri modellemeye yönelik, matematiksel açıdan titiz ve cebirsel olarak ifade gücü yüksek bir kuramsal çerçeve sunmaktadır. Bu biçimsel yapının merkezinde, esnek kümeler evrenine zengin ve ince yapılandırılmış bir iç cebirsel topoloji kazandıran kapsamlı bir cebirsel işlemler dizisi ve ikili çarpım yapıları yer almaktadır. Bu cebirsel bağlam içerisinde, parametre kümeleri grup kuramsal bir yapıya sahip olan esnek kümeler üzerinde tanımlı esnek birleşim–gama çarpımı olarak adlandırılan yeni bir ikili işlem tanıtılmakta ve biçimsel olarak araştırılmaktadır. Söz konusu işlem, esnek altkümelik ve esnek eşitlik kavramlarının genelleştirilmiş biçimleriyle uyumluluğu güvence altına alan aksiyomatik bir çerçeve içinde titizlikle inşa edilmiştir. Bu işlemin temel özelliklerini belirlemek amacıyla sistematik bir cebirsel analiz gerçekleştirilmiştir. Yapılan çözümleme kapsamında kapalılık, birleşmelilik, değişmeli olma, idempotentlik ve diğer esnek küme işlemleri üzerine dağılma gibi temel yapısal özelliklerin yanı sıra, birim ve yutan elemanlara göre davranışı da ayrıntılı biçimde ele alınmıştır. Ayrıca, işlemin boş ve evrensel esnek kümelerle olan etkileşimi de kesin biçimde tanımlanmıştır. Kuramsal bulgular, bu işlemin grup yapılandırmalı parametre kümeleri tarafından dayatılan cebirsel kısıtlamaları karşıladığını ve esnek küme uzayı üzerinde yapısal açıdan tutarlı ve iyi örgütlenmiş bir cebirsel sistem oluşturduğunu göstermektedir. Bu çalışmadan iki temel katkı öne çıkmaktadır: Birincisi, esnek birleşim–gama çarpımının teoriye dâhil edilmesi, esnek küme teorisinin içsel işlemsel uyumunu artırmakta ve onu biçimsel olarak tutarlı, aksiyomlarla yönetilen bir cebirsel ortam içerisine yerleştirmektedir. İkincisi ise, grup yapısına sahip parametre kümeleri tarafından indekslenen esnek kümelerin, uygun şekilde tanımlanmış esnek işlemler aracılığıyla klasik grup kuramı sistemlerinin aksiyomatik davranışlarını yansıttığı genelleştirilmiş bir esnek grup teorisinin kavramsal ve yapısal temelini oluşturmaktadır. Bu çalışma, esnek küme teorisinin cebirsel olarak pekiştirilmesi ve kuramsal olarak genelleştirilmesi yönünde önemli bir ilerlemeyi temsil etmektedir.

Keywords

Esnek kümeler , Esnek altkümeler , Esnek eşitlikler , Esnek birleşim-gama çarpımı

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