Algebraic Structures using Subsets

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a semiring or a semifield or a ring or a field; if we g ive the inherited operations of the sem igroup or a group or a sem iring or a semifield or a ring or a field respectively; the resulting structure is always a semigroup or a semiring or a semifield only. They can never get the structure of a group or a field or a ring. We call these new algebraic structures as subset semigroups or subset semirings or subset semifields. This method gives us inf inite num ber of finite noncommutative semirings. Using these subset semirings, subset semifields and subset semigroups we can define subset ideal topological spaces and subset set ideal topol ogical spaces. Further us ing subset semirings an d subset semifields we can build new subset topological set ideal spaces which may not be a commutative topological space. This i nnovative methods gives non commutative new set ideal topological spaces provided the under lying structure used by us is a noncommutative semiring or a noncommutative ring