Subset Non Associative Semirings

In this book for the first time we introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this specific topic is still rare. Authors have recently introduced the notion of subset algebraic structures. The maximum algebraic structure enjoyed by subsets with two binary operations is just a semifield and semiring, even if a ring or a field is used. In case semigroups or groups are used still the algebraic structure of the subset is only a semigroup. To construct a subset non associative semiring we use either a non associative ring or a non associative semiring. This study is innovative and interesting. We construct subset non associative semirings using groupoids. We define notions like Smarandache non associative subset semirings, sub structures in them and study their properties. Finite subset non associative semirings are constructed using the groupoid lattice LP where L is a finite distributive lattice and P is a groupoid of finite order. Using also loop lattice we can have finite subset non associative semirings. When in the place of the lattice a semiring is used we get non associative semirings and the collection all subsets of them form a subset non associative semiring