Smarandache Special Definite Algebraic Structures

TL;DR

This work introduces Smarandache special definite algebraic structures, which are strong algebraic structures containing proper subsets that are weaker algebraic structures, expanding the understanding of algebraic hierarchy.

Contribution

It defines and explores new classes of algebraic structures with proper subsets being weaker structures, providing properties and numerous problems for further research.

Findings

Smarandache structures exist only in infinite algebraic systems.

Several properties of these structures are derived in groups, fields, rings, and vector spaces.

Over 200 problems are proposed for further exploration.

Abstract

In this book, we introduce the notion of Smarandache special definite algebraic structures. We can also call them equivalently as Smarandache definite special algebraic structures. These new structures are defined as those strong algebraic structures which have in them a proper subset which is a weak algebraic structure. For instance, the existence of a semigroup in a group or a semifield in a field or a semiring in a ring. It is interesting to note that these concepts cannot be defined when the algebraic structure has finite cardinality i.e., when the algebraic structure has finite number of elements in it. This book has four chapters. Chapter one is introductory in nature. In chapter two, the notion of Smarandache special definite groups and Smarandache special definite fields are introduced and several interesting properties are derived. The notion of Smarandache definite special rings, vector spaces and linear algebras are introduced and analyzed in chapter three. The final chapter suggests over 200 problems.