In this paper, we define soft intersection semigroups, soft intersection left (right, two-sided) ideals and bi-ideals of semigroups, give their properties and interrelations and we characterize regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups in terms of these ideals.

The soft set theory developed by Molodtsov has been applied both theoretically and practically in many fields. It is a useful piece of mathematics for handling uncertainty. Numerous variations of soft set operations have been described and used since its introduction. In this paper, we define a new soft set operation, called complementary soft binary piecewise intersection operation, a unique soft set operation, and examine its basic algebraic properties. Additionally, we aim to contribute to the literature on soft sets by illuminating the relationships between this new soft set operation and other kinds of soft set operations by researching the distribution of this new soft set operation over other soft set operations. Moreover, we prove that the set of all the soft sets with a fixed parameter set together with the complementary soft binary piecewise intersection operation and the soft binary piecewise union operation is a zero-symmetric near-semiring and hemiring.

Molodtsov introduced the theory of soft sets, which can be seen as an effective mathematical tool to deal with uncertainties, since it is free from the difficulties that the usual theoretical approaches have troubled. In this paper, we apply the definitions proposed by Ali et al. [M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547-1553] to the concept of soft near- rings and substructures of soft near-rings, proposed by Atag?n and Sezgin [A. O. Atag?n and A. Sezgin, Soft Near-rings, submitted] and show them with illustrating examples. Moreover, we investigate the properties of idealistic soft near-rings with respect to the near-ring mappings and we show that the structure is preserved under the near-ring epimorphisms. Main purpose of this paper is to extend the study of soft near-rings from a theoretical aspect.

Soft set theory is a theory of dealing with uncertainty. Since its inception, many kinds of soft set operations are defined and used in various types. In this paper, a new kind of soft set operation called, complementary soft binary piecewise difference operation is defined and its basic properties are investigated. We obtain many striking analogous fact between difference operation in classical theory and complementary soft binary piecewise difference operation in soft set theory. Also, by obtaining the relationships between this new soft set operation and all other types of soft set operations, we aim to contribute to the soft set literature with the help of examing the distribution rules.

Molodtsov tarafından geliştirilen esnek küme teorisi hem teorik hem de pratik olarak birçok alanda uygulanmıştır. Belirsizliği ele almak için yararlı bir matematiksel araçtır. Ortaya atıldığından bu yana çok sayıda esnek küme işlemi varyasyonu tanımlanmış ve kullanılmıştır. Bu çalışmada, esnek ikili parçalı simetrik fark işlemi adı verilen yeni bir esnek küme işlemi tanımlanıp, özellikleri klasik küme teorisinde var olan simetrik fark işleminin temel cebirsel özellikleri ile karşılaştırmalı olarak ele alınmış ve incelenmiştir. Ayrıca, esnek ikili parçalı simetrik fark işlemi ve kısıtlanmıs kesisim işlemleri ile birlikte sabit parametreye sahip tüm esnek kümelerin oluşturduğu cebirsel yapının, birimli ve değişmeli bir hemiring ve ayrıca Boole halkası olduğu gösterilmiştir.

In 1999, Molodtsov introduced Soft Set Theory as a mathematical tool to deal with uncertainty. It has been applied to many fields both as theoretical and application aspects. Since 1999, different kinds of soft set operations have been defined and used in various types. In this paper, we define a new kind of soft set operation called, “complementary soft binary piecewise lambda operation” and we handle its basic algebraic properties. Also, it is intended to contribute to the literature of soft set by gaining the relationships between this new soft set operation and some other types of soft set operations via examining the distribution of complementary soft binary piecewise lambda operation over extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations in order to inspire to obtain the algebaric structures of soft sets and some new decision making methods.