Soft Intersection-lambda Product of Groups

Year 2025, Volume: 4 Issue: 1, 1 - 15, 31.12.2025

İbrahim Durak , Aslıhan Sezgin

Abstract

Soft set theory constitutes a mathematically rigorous and algebraically expressive formalism for modeling systems permeated by epistemic indeterminacy, vagueness, and parameter-dependent variability—characteristics that are endemic to foundational problems in decision theory, engineering, economics, and the information sciences. Central to this framework is a broad spectrum of algebraic operations and binary product constructions that collectively impart a deep and intricate internal structure to the universe of soft sets, capable of encoding complex parametric interdependencies with high fidelity. Within this context, we propose and systematically examine a novel product, termed the soft intersection-lambda product, defined over soft sets whose parameter sets are endowed with a group-theoretic structure. The operation is rigorously axiomatized to ensure formal coherence with generalized notions of soft subsethood and soft equality, thereby preserving the algebraic integrity of the underlying system. A detailed algebraic analysis is conducted to investigate fundamental structural properties of the operation—including closure, associativity, commutativity, idempotency, the presence of identity and absorbing elements, and distributy over other soft set operations—as well as its interactions with the null and absolute soft sets. Furthermore, the proposed product is analytically compared with previously established soft binary operations within the taxonomy of soft subset classifications, yielding refined insights into their relative expressive power and mutual algebraic compatibility. Theoretical findings confirm that the product not only adheres to the structural constraints imposed by the group-parameterized domain but also engenders a formally consistent and well-behaved algebraic system on the collection of soft sets. Two principal algebraic implications emerge from this investigation: (i) the integration of the soft intersection-lambda product enhances the internal operational cohesion of soft set theory by embedding it within an axiomatically sound and operation-preserving environment, and (ii) the proposed product serves as a conceptual cornerstone for the development of a generalized soft group theory, wherein soft sets defined over group-structured parameter spaces emulate the axiomatic behavior of classical group-theoretic constructs under newly defined soft operations. Given that the algebraic maturation of soft set theory hinges upon the rigorous formulation of operations that satisfy semantically and structurally significant axioms, the present work represents a substantial advancement in the algebraic unification and generalization of the field. Beyond its theoretical import, the proposed operation offers practical utility in the construction of abstract algebra-based soft computational models, with far-reaching applications in multi-criteria decision-making systems, algebraically guided classification schemes, and uncertainty-aware data analysis frameworks over group-parameterized semantic spaces. Accordingly, the algebraic architecture developed herein significantly expands the foundational landscape of soft set theory and consolidates its relevance across both pure and applied mathematical domains.

Keywords

Soft sets , soft subsets , soft equalities , soft intersection-lambda product

Grupların Esnek Kesişim-lamda Çarpımı

Abstract

Esnek küme teorisi, epistemik belirsizlik, muğlaklık ve parametreye bağlı değişkenlik gibi özelliklerle kuşatılmış sistemleri modellemek için matematiksel olarak titiz ve cebirsel olarak ifade gücü yüksek bir formel yapı sunar. Bu özellikler, karar teorisi, mühendislik, ekonomi ve bilgi bilimlerindeki temel problemlerde yaygın olarak görülmektedir. Bu çerçevenin merkezinde, esnek kümeler evrenine derin ve karmaşık bir iç yapı kazandıran, karmaşık parametreler arası bağımlılıkları yüksek doğrulukla kodlayabilen çok çeşitli cebirsel işlemler ve ikili çarpım yapıları yer alır.Bu bağlamda, parametre kümeleri grup kuramı (grup teorisi) yapısıyla donatılmış esnek kümeler üzerinde tanımlanan ve esnek kesişim-lamda çarpımı olarak adlandırılan yeni bir çarpım işlemi önerilmekte ve sistematik olarak incelenmektedir. Bu işlem, esnek altkümelik ve esnek eşitlik kavramlarının genelleştirilmiş biçimleriyle biçimsel tutarlılığı sağlamak üzere aksiyomatik olarak tanımlanmış olup, altında yatan sistemin cebirsel bütünlüğünü koruyacak şekilde yapılandırılmıştır. Önerilen işlemin temel yapısal özelliklerini araştırmak amacıyla kapsamlı bir cebirsel analiz gerçekleştirilmiştir; bu analizde kapalılık, birleşme, değişme, idempotentlik, birim ve yutan elemanların varlığı ile diğer esnek küme işlemleri üzerindeki dağılım gibi özellikler ve bu işlemin boş ve evrensel (mutlak) esnek kümelerle olan etkileşimleri ele alınmıştır. Ayrıca, önerilen çarpım işlemi daha önce literatürde tanımlanmış esnek ikili işlemlerle karşılaştırılarak, ifade gücü ve cebirsel uyumluluk açısından derinlemesine analiz edilmiştir. Teorik bulgular, bu işlemin yalnızca grup-parametreli tanım alanının yapısal kısıtlamalarına uyum sağlamakla kalmayıp, aynı zamanda esnek kümeler kümesi üzerinde biçimsel olarak tutarlı ve iyi tanımlı bir cebirsel sistem oluşturduğunu da doğrulamaktadır. Bu çalışmadan iki temel cebirsel sonuç elde edilmiştir: (I) Esnek kesişim-lambda çarpımının entegrasyonu, esnek küme teorisinin içsel işlemsel tutarlılığını artırmakta ve onu aksiyomatik olarak sağlam, işlem koruyucu bir yapı içerisine yerleştirmektedir. (II) Önerilen çarpım, grup yapılı parametre alanları üzerinde tanımlı esnek kümelerin, klasik grup teorisindeki yapıları taklit eden yeni tanımlı esnek işlemler altında davranış göstermesini sağlayarak, genelleştirilmiş bir esnek grup teorisinin gelişimi için kavramsal bir temel sunmaktadır. Esnek küme teorisinin cebirsel olarak olgunlaşması, anlamsal ve yapısal olarak anlamlı aksiyomları karşılayan işlemlerin titizlikle formüle edilmesine bağlı olduğundan, bu çalışma, alanın cebirsel birleşimi ve genelleştirilmesi yönünde önemli bir ilerlemeyi temsil etmektedir. Teorik öneminin ötesinde, önerilen işlem, çok kriterli karar verme sistemlerinde, cebirsel olarak yönlendirilmiş sınıflandırma şemalarında ve grup-parametreli anlamsal uzaylarda belirsizlik duyarlı veri analiz çerçevelerinde kullanılabilecek soyut cebire dayalı esnek hesaplama modellerinin inşasında pratik bir yarar sunmaktadır. Böylece, burada geliştirilen cebirsel yapı, esnek küme teorisinin temellerini anlamlı şekilde genişletmekte ve hem teorik hem de uygulamalı matematik alanlarında önemini pekiştirmektedir.

Keywords

Esnek kümeler , esnek alt kümeler , esnek eşitlikler , esnek kesişim-lamda işlemi

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