Soft Loops

Foundation of soft sets was laid by Molodtsov [14] in 1999 in order to introduce a mathematical tool for modeling vagueness and uncertainty arising in computer sciences, natural sciences, social sciences, medical and many other fields. This is one of mathematical concepts like fuzzy sets [21], generalized fuzzy sets [6] and rough sets [16] given in order to overcome the difficulties in dealing with uncertainties arising in above mentioned fields and daily life, which were hard to handle with, in classical set theory. A soft set is a parameterized family of subsets of a universal set. This can be considered as a neighborhood system, and is a generalized case of context-dependent fuzzy set. There does not occur the problem of setting the membership function, among other related problems in soft sets. This makes it very convenient and easy to apply in practice. Because of this friendly behavior, soft sets have various applications within and outside mathematics. For basic notions and the applications of soft sets, we incite to read [1, 4, 5, 10, 12, 13, 14, 15]. The study of non-associative structures is justified since two out of four basic binary operations are non-associative. A loop Q is a non-associative algebraic structure, which is defined as usual, a groupoid with binary operation ’·’ having an element e such that e · x = x · e = x, for all x ∈ Q and in the equation x · y = z any two of x, y, z uniquely determine the third. The study of loops started in 1920’s and these were introduced for the first time in 1930’s [19]. The theory of loops has its roots in geometry, algebra and combinatorics. This can be found in non-associative products in algebra, in combinatorics this is present in latin squares of particular form and in geometry it has connection with the analysis of web structures [18]. A detailed study of theory of the loops can be found in [2, 3, 7, 8, 9, 18]. In [1], authors defined soft groups and gave fundamental results in soft group theory as a generalization of fuzzy groups [20]. They showed that a fuzzy group is a special case of soft group. This gave a new dimension to soft set theory and involved algebra in it. This work was extended by Aslam and Qurashi in [5], discussing various structural properties of soft groups their substructures and structure preserving mappings.