Smarandache near-rings

The main concern of this book is the study of Smarandache analogue properties of near-rings and Smarandache near-rings; so it does not promise to cover all concepts or the proofs of all results

Smarandache Near-rings

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B contained in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Near-ring is a non-empty set N together with two binary operations '+' and '.' such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c belonging to N we have (a + b) . c = a . c + b . c A Near-field is a non-empty set P together with two binary operations '+' and '.' such that (P, +) is a group (not-necessarily abelian), {P\{0}, .) is a group. For all a, b, c belonging to P we have (a + b) . c = a . c + b . c A Smarandache Near-ring is a near-ring N which has a proper subset P contained in N, where P is a near-field (with respect to the same binary operations on N).Comment: 200 pages, 50 tables, 20 figure

Characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed

A Note on One Sided and Two Sided PO-Ternary Ideals in PO-Ternary Semiring

In this paper the term, left(lateral, right and two sided) PO-ternary ideal, maximal left(lateral, right and two sided) PO-ternary ideal, left (lateral, right and two sided) PO-ternary ideal of T generated by a set A, principal left (lateral, right and two sided) PO-ternary ideal generated by an element a left (lateral, right and two sided) simple PO-ternary semiring are introduced. It is proved that (1) the non-empty intersection of any two left (lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left (lateral, right and two sided) PO-ternary ideal of T. (2) non-empty intersection of any family of left (lateral, right and two sided) PO-ternary ideals of a POternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (3) the union of any left PO-ternary ideals of a PO-ternary semiring T is a left PO-ternary ideal of T. (4) the union of any family of left(lateral, right and two sided) PO-ternary ideals of a PO-ternary semiring T is a left(lateral, right and two sided) PO-ternary ideal of T. (5) The left (lateral, right and two sided) PO-ternary ideal of a PO-ternary semiring T generated by a non-empty subset A is the intersection of all left(lateral, right and two sided) PO-ternary ideals of T containing A. (6) If T is a PO-ternary semiring and a∈T then L(a) =  (Te Tea + na] = (Te Tea U na] (M(a) = (Tea Te + TeTe aTeTe + na] = ( Tea Te U TeTe aTeTe U na], R (a) = (aTe Te + na ] = (aTe Te U na] and T(a) = (Te Tea + aTe Te + TeTe aTeTe +na] = (Te Tea UaTe Te UTeTe aTeTe Una]). (7) A PO-ternary semiring T is a left(lateral, right) simple PO-ternary semiring if and only if (TTa] = T ((TaT U TTaTT] = T, (aTT] = T) for all a∈T

Neutrosophic Rings

This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings,neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem.Comment: 154 page

Characterizations of Regular Ordered Semirings by Ordered Quasi-Ideals

We introduce the notion of an ordered quasi-ideal of an ordered semiring and show that ordered quasi-ideals and ordered bi-ideals coincide in regular ordered semirings. Then we give characterizations of regular ordered semirings, regular ordered duo-semirings, and left (right) regular ordered semirings by their ordered quasi-ideals

Fuzzy Quasi-ideals of Ordered Semigroups

In this paper, we characterize ordered semigroups in terms of fuzzy quasi-ideals. We characterize left simple, right simple and completely regular ordered semigroups in terms of fuzzy quasi-ideals. We define semiprime fuzzy quasi-ideal of ordered semigroups and characterize completely regular ordered semigroup in terms of semiprime fuzzy quasi-ideals. We also study the decomposition of left and right simple ordered semigroups having the property a a 2 for all a 2 S, by means of fuzzy quasi-ideals

A STUDY ON STRUCTURE OF PO-TERNARY SEMIRINGS

This paper is divided into two sections. In section 1, the notion of a PO-ternary semiring was introduced and examples are given.  Further the terms commutative PO-ternary semiring , quasi commutative PO-ternary semiring, normal PO-ternary semiring, left pseudo commutative PO-ternary semiring, lateral  pseudo commutative PO-ternary semiring, right pseudo commutative PO-ternary semiring and pseudo commutative PO-ternary semiring are introduced and characterized them.  Further the terms left singular, right singular and singular with respect to addition and left singular, right singular, lateral singular, singular with respect to ternary multiplication and two sided singular are introduced and made a study on them. In section 2, the terms; PO-ternary subsemiring, PO-ternary subsemiring of T generated by a set A, cyclic PO-ternary subsemiring and cyclic PO-ternary semiring are introduced.  It is proved that T be a PO-ternary semiring and A be a non-empty subset of T.  Then (A) = {a1a2....an-1;n ϵ N, a1,a2....an Ïµ A } is a smallest PO-ternary subsemiring of T.  Let T be a PO-ternary semiring and A be a non-empty subset of T.  <A> = the intersection of all PO-ternary subsemirings of  T containing A