Soft Union-lambda Product of Groups

Year 2025, Volume: 9 Issue: 2, 64 - 82, 30.09.2025

İbrahim Durak , Aslıhan Sezgin

https://doi.org/10.30516/bilgesci.1727597

Abstract

Soft set theory provides a logically sound and algebraically rich framework for modeling systems characterized by ambiguity, epistemic uncertainty, and parameter-dependent variability. This study introduces the soft union–lambda product, a novel binary operation defined over soft sets whose parameter spaces are endowed with an intrinsic group-theoretic structure. Developed within a rigorously formulated axiomatic framework, the opera-tion is shown to be fully compatible with generalized notions of soft subsethood and soft equality. A thorough algebraic investigation is undertaken to establish the fundamental structural properties of the opera-tion—including closure, associativity, commutativity, idempotency, and its distributivity over other soft set operations—alongside a precise characterization of its behavior with respect to identity, absorbing, null, and absolute soft sets. The results demonstrate that the soft union–lambda product adheres to all algebraic con-straints dictated by group-parameterized domains, thereby inducing a consistent and internally cohesive alge-braic structure on the universe of soft sets. Beyond its foundational contributions, the proposed operation sig-nificantly expands the formal toolkit of soft set theory and sets the stage for the development of a generalized soft group theory. Furthermore, its formal alignment with key relational structures such as soft equality and soft inclusion highlights its potential applicability across a diverse array of analytical contexts, including abstract algebraic modeling, uncertainty-aware classification, and multi-criteria decision analysis. Accordingly, the findings of this study offer both profound theoretical advancements and concrete pathways for practical im-plementation.

Keywords

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

Grupların esnek birleşim-lamda çarpımı

Abstract

Esnek küme teorisi, belirsizlik, epistemik kararsızlık ve parametreye bağlı değişkenlik gibi niteliklerle tanımlanan sistemlerin modellenmesinde mantıksal olarak tutarlı ve cebirsel açıdan zengin bir çerçeve sunar. Bu çalışma, parametre kümesi grup olan yapılarla donatılmış esnek kümeler üzerinde tanımlanan yeni bir ikili işlem olan esnek birleşim–lambda çarpımı’nı tanıtmaktadır. Sıkı biçimde yapılandırılmış aksiyomatik bir çerçevede geliştirilen bu işlem, genelleştirilmiş esnek altkümelik ve esnek eşitlik kavramlarıyla tam uyum içinde olduğu gösterilmiştir. İşlemin temel yapısal özelliklerini—kapalılık, birleşmelilik, değişmelilik, idempotentlik ve diğer esnek küme işlemleri üzerindeki dağılım özelliği dahil olmak üzere—ortaya koymak amacıyla kapsamlı bir cebirsel inceleme gerçekleştirilmiştir. Ayrıca, bu işlemin birim, yutan, boş ve mutlak esnek kümelerle olan ilişkisi ayrıntılı biçimde analiz edilmiştir. Elde edilen bulgular, esnek birleşim–lambda çarpımının grup-yapılı parametre alanlarının dayattığı tüm cebirsel kısıtlamalara uyduğunu ve böylece esnek kümeler evreni üzerinde tutarlı ve içsel olarak bütünlüklü bir cebirsel yapı oluşturduğunu göstermektedir. Bu işlemin kuramsal açıdan sunduğu temel katkıların ötesinde, önerilen yaklaşım esnek küme teorisinin biçimsel araç setini önemli ölçüde genişletmekte ve genelleştirilmiş bir esnek grup teorisinin gelişimine zemin hazırlamaktadır. Ayrıca, esnek eşitlik ve esnek içerme gibi temel ilişkisel yapılarla biçimsel uyumu, bu işlemin soyut cebirsel modelleme, belirsizlik duyarlı sınıflandırma ve çok ölçütlü karar analizleri gibi çeşitli analitik bağlamlarda uygulanabilirliğini öne çıkarmaktadır. Bu doğrultuda, çalışmanın bulguları hem derin kuramsal ilerlemeler hem de pratik uygulamalara yönelik somut yollar sunmaktadır.

Keywords

Esnek kümeler , Esnek alt kümeler , Esnek eşitlikelr , Esnek birleşim-lamda çarpımı

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