Esnek Kümelerin Kısıtlanmış ve Genişletilmiş Kesişim İşlemleri Üzerine Kapsamlı Bir Çalışma

Öz

Esnek küme teorisi, Molodtsov tarafından ortaya atıldığından beri belirsizlikle ilgili problemleri ele almak ve belirsizliği modellemek için devrim niteliğinde bir yaklaşım olarak öne çıkmıştır. Teorinin temel kavramı olan esnek küme işlemleri kavramı, teorideki teorik ve pratik ilerlemeler için temel teşkil etmiş, bu nedenle esnek küme işlemlerinin cebirsel özelliklerini türetmek ve esnek küme işlemleriyle ilişkili esnek kümelerin cebirsel yapısını incelemek araştırmacıların ilgisini sürekli çekmiştir. Esnek küme teorisinde, şimdiye kadar aralarında bazı farklılıklar bulunan ve bazıları esasen kullanışlı ve işlevsel olmadıkları için artık kullanımı tercih edilmeyen birçok esnek kesişim işlemi tanımlanmıştır. Kısıtlı kesişim tanımı literatürde yaygın olarak kabul görse ve çalışmalarda kullanılsa da, esnek kümelerin parametre kümelerinin ayrık olabileceği bazı durumlar göz ardı edildiğinden, teoremlerdeki tüm durumlar ilgili ispatlarda dikkate alınmadığından, bu işlemin kullanıldığı veya özelliklerinin araştırıldığı çalışmalarda yanlışlık veya eksikliğe neden olmaktadır. Bu bağlamda, mevcut literatürde, doğru tanımlanmış kısıtlı kesişim işlemi ile genişletilmiş kesişim işleminin doğru özellikleri ve dağılımları ve bu esnek küme işlemleriyle ilişkili doğru cebirsel yapılar hakkında kapsamlı bir çalışmanın kritik bir eksikliği vardır. Bu çalışmada, öncelikle kısıtlı kesişim tanımının sunumundaki eksiklikleri düzelterek ve revize ederek bu önemli boşluğu doldurmayı amaçlıyoruz. Ayrıca, bu işlemlerle ilgili birçok makalede, birkaç teorem ispatları olmadan sunulmuş veya ispatlarda bazı hatalı kısımlar bulunmaktadır. Bu çalışmada, fonksiyon eşitliğine dayalı tüm ispatlar düzenli olarak sağlanmış ve ayrıca, esnek alt küme kavramı ile kısıtlı ve genişletilmiş kesişim işlemleri arasındaki ilişkiler ilk kez ayrıntılı ispatlarıyla sunulmuştur. Dahası, bu işlemlerin klasik küme teorisindeki kesişim işleminin analojisi ve karşılığı olarak birçok yeni özelliği elde edilmiştir. Ayrıca, işlemlerin tüm özellikleri ve diğer esnek küme işlemleri üzerindeki dağılımları, işlemlerin hem evrensel küme üzerindeki esnek kümeler kümesinde hem de sabit parametre kümesinde ayrı ayrı ve diğer esnek küme işlemleriyle birlikte oluşturdukları doğru cebirsel yapıları belirlemek için kapsamlı bir şekilde araştırılmıştır. Kısıtlanmış/genişletilmiş kesişim işlemlerinin, diğer esnek küme işlemleriyle birleştirildiğinde, monoid, sınırlı yarı kafes, yarı halka, hemiring, sınırlı dağıtımlı kafes, Bool cebiri, De Morgan Cebri, Kleene Cebri, Stone cebiri ve MV-cebri gibi birkaç önemli cebirsel yapı oluşturduğu ayrıntılı açıklamalarla gösterilmiştir. Bu bağlamda, bu genel çalışma, kısıtlı kesişim ve genişletilmiş kesişim konusunda literatürde bugüne kadar yapılmış en kapsamlı analizi temsil etmektedir; çünkü bu konu üzerinde yapılmış tüm önemli araştırmaları, düzeltilmiş teoremleri ve bunların ispatlarını kapsamakta, böylece literatürdeki önemli boşluğu doldurarak teoriyi ilerletmekte, bu popüler teoriye yeni başlayanlar için bir rehber görevi görmekte ve ayrıca esnek kümeler üzerine yapılacak gelecekteki çalışmalara ışık tutmaktadır.

Anahtar Kelimeler

• Esnek kümeler
• Esnek küme işlemleri
• Kısıtlanmıs kesişim işlemi
• Genişletilmiş kesişim işlemi

A Comprehensive Study on Restricted and Extended Intersection Operations of Soft Sets

Öz

Soft set theory has gained prominence as a revolutionary approach for handling uncertainty-related problems and modeling uncertainty since it was proposed by Molodtsov. The concept of soft set operations, which is the major notion for the theory, has served as the foundation for theoretical and practical advances in the theory, therefore deriving the algebraic properties of the soft set operations and studying the algebraic structure of soft sets associated with soft set operations have attracted the researchers’ interest continuously. In the theory of soft set, many soft intersection operations have been defined up to now among which there are some differences, and some of which are no longer preferred for use as they are essentially not useful and functional. Although the definition of restricted intersection is widely accepted in the literature and used in the studies, it is still incomplete with its current form suffering from certain cases where the parameter sets of the soft sets may be disjoint is ignored, thus all the circumstances in the theorems are not considered in the related proofs causing to the incorrectness or deficiency in the studies where this operation is used or its properties are investigated. In this regard, in the existing literature, there is a critical lack of comprehensive study on the correct defined restricted intersection operation together with extended intersection including their correct properties and distributions and the correct algebraic structures assoiciated with these soft set operations. In this study, we primarly intend to fill this crucial gap by first correcting the deficiencies in the presentation of the definition of restiricted intersection and revising it. Moreover, in many papers related to these operation, several theorems were presented without their proofs, or there were some incorrect parts in the proofs. In this study, all the proofs based on the function-equality are regularly provided and besides, the relationships between the concept of soft subset and restricted and extended intersection operations are presented for the first time with their detailed proofs. Furhermore, we obtain many new properties of these operations as analogy and counterpart of intersection operation in classical set theory. Moreover, the operations’ full properties and distributions over other soft set operations are throughly investigated to determine the correct algebraic structures the operations form individually and in combination with other soft set operations both in the set of soft sets over the universe and with a fixed parameter set. We demonstrate that the restricted/extended intersection operations, when combined with other kinds of soft set operations, form several significant algebraic structures, such as monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, Kleene Algebra, Stone algebra and MV-algebra but with deteailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted intersection and extended intersection in the literature to date as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by filling the significant gap in the literature, acting as a guide for the beginners of this popular theory, and besides shedding light on the future studies on soft sets.

Anahtar Kelimeler

• Soft sets
• soft set operations
• restricted intersection operation
• extended intersection operation

Kaynakça

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