Smarandache Special Definite Algebraic Structures

Author : W. B. Vasantha Kandasamy
Publisher : Infinite Study
Page : 141 pages
File Size : 46,6 Mb
Release : 2009-01-01
Category : Mathematics
ISBN : 9781599730851

Get Book

Smarandache Special Definite Algebraic Structures by W. B. Vasantha Kandasamy Pdf

We study these new Smarandache algebraic structures, which are defined as structures which have a proper subset which has a weak structure.A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure.By proper subset of a set S, we mean a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure.A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure.

Author : W. B. Vasantha Kandasamy,Florentin Smarandache
Publisher : Infinite Study
Page : 172 pages
File Size : 41,5 Mb
Release : 2011
Category : Mathematics
ISBN : 9781599731353

Get Book

Algebraic Structures Using Natural Class of Intervals by W. B. Vasantha Kandasamy,Florentin Smarandache Pdf

Author : W. B. Vasantha Kandasamy,Florentin Smarandache
Publisher : Infinite Study
Page : 209 pages
File Size : 46,7 Mb
Release : 2005-01-01
Category : Mathematics
ISBN : 9781931233057

Get Book

N-Algebraic Structures by W. B. Vasantha Kandasamy,Florentin Smarandache Pdf

In this book, for the first time we introduce the notions of N-groups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure. The book is organized into six chapters. The first chapter gives the basic notions of S-semigroups, S-groupoids and S-loops thereby making the book self-contained. Chapter two introduces N-groups and their Smarandache analogues. In chapter three, N-loops and Smarandache N-loops are introduced and analyzed. Chapter four defines N-groupoids and S-N-groupoids. Since the N-semigroup structures are sandwiched between groups and groupoids, the study can be carried out without any difficulty. Mixed N-algebraic structures and S-mixed algebraic structures are given in chapter five. Some problems are suggested in chapter six. It is pertinent to mention that several exercises and problems (Some in the form of proof to the theorems are given in all the chapters.) A reader who attempts to solve them will certainly gain a sound knowledge about these concepts. We have given 50 problems for the reader to solve in chapter 6. The main aim of this book is to introduce new concepts and explain them with examples there by encouraging young mathematics to pursue research in this direction. Several theorems based on the definition can be easily proved with simple modification. Innovative readers can take up that job. Also these notions find their applications in automaton theory and coloring problems. The N-semigroups and N-automaton can be applied to construct finite machines, which can perform multitasks, so their capability would be much higher than the usual automaton of finite machines constructed. We have suggested a list of references for further reading.

Author : W. B. Vasantha Kandasamy, Florentin Smarandache
Publisher : Infinite Study
Page : 168 pages
File Size : 43,7 Mb
Release : 2013
Category : Mathematics
ISBN : 9781599732121

Get Book

Set Theoretic Approach to Algebraic Structures in Mathematics - A Revelation by W. B. Vasantha Kandasamy, Florentin Smarandache Pdf

Author : Florentin Smarandache
Publisher : Infinite Study
Page : 11 pages
File Size : 46,6 Mb
Release : 2020-03-12
Category : Mathematics
ISBN : 8210379456XXX

Get Book

NeutroAlgebra is a Generalization of Partial Algebra by Florentin Smarandache Pdf

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to and , and one corresponding to neutral (indeterminate) (also denoted ) between the opposites}, which may or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation).

Author : Arsham Borumand Saeid
Publisher : Infinite Study
Page : 65 pages
File Size : 41,9 Mb
Release : 2024-05-23
Category : Electronic
ISBN : 9781599732411

Get Book

Smarandache BE-Algebras by Arsham Borumand Saeid Pdf

v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:8.0pt; mso-para-margin-left:0in; line-height:107%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin;} There are three types of Smarandache Algebraic Structures: 1. A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure. A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure. A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure. By proper subset of a set S, one understands a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any. Having two structures {u} and {v} defined by the same operations, one says that structure {u} is stronger than structure {v}, i.e. {u} > {v}, if the operations of {u} satisfy more axioms than the operations of {v}. Each one of the first two structure types is then generalized from a 2-level (the sets P ⊂ S and their corresponding strong structure {w1}>{w0}, respectively their weak structure {w1}<{w0}) to an n-level (the sets Pn-1 ⊂ Pn-2 ⊂ … ⊂ P2 ⊂ P1 ⊂ S and their corresponding strong structure {wn-1} > {wn-2} > … > {w2} > {w1} > {w0}, or respectively their weak structure {wn-1} < {wn-2} < … < {w2} < {w1} < {w0}). Similarly for the third structure type, whose generalization is a combination of the previous two structures at the n-level. A Smarandache Weak BE-Algebra X is a BE-algebra in which there exists a proper subset Q such that 1 Q, |Q| ≥ 2, and Q is a CI-algebra. And a Smarandache Strong CI-Algebra X is a CI-algebra X in which there exists a proper subset Q such that 1 Q, |Q| ≥ 2, and Q is a BE-algebra. The book elaborates a recollection of the BE/CI-algebras, then introduces these last two particular structures and studies their properties.

Author : W. B. Vasantha Kandasamy, Florentin Smarandache
Publisher : Infinite Study
Page : 199 pages
File Size : 41,8 Mb
Release : 2012
Category : Algebra, Boolean
ISBN : 9781599732169

Get Book

Algebraic Structures Using Subsets by W. B. Vasantha Kandasamy, Florentin Smarandache Pdf

"[The] study of algebraic structures using subsets [was] started by George Boole. After the invention of Boolean algebra, subsets are not used in building any algebraic structures. In this book we develop algebraic structures using subsets of a set or a group, or a semiring, or a ring, and get algebraic structures. Using group or semigroup, we only get subset semigroups. Using ring or semiring, we get only subset semirings. By this method, we get [an] infinite number of non-commutative semirings of finite order. We build subset semivector spaces, [and] describe and develop several interesting properties about them."--

Author : Florentin Smarandache
Publisher : Infinite Study
Page : 16 pages
File Size : 50,7 Mb
Release : 2024-05-23
Category : Mathematics
ISBN : 8210379456XXX

Get Book

Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited) by Florentin Smarandache Pdf

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined. Again, in all classical algebraic structures, the Axioms (Associativity, Commutativity, etc.) defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations in science and in any domain of knowledge when an Axiom defined on a set may be only partially-true (and partially-false), that we call NeutroAxiom, or totally false that we call AntiAxiom. Therefore, we open for the first time in 2019 new fields of research called NeutroStructures and AntiStructures respectively.

Author : FLORENTIN SMARANDACHE
Publisher : Infinite Study
Page : 5 pages
File Size : 47,8 Mb
Release : 2024-05-23
Category : Electronic
ISBN : 8210379456XXX

Get Book

Special Algebraic Structures by FLORENTIN SMARANDACHE Pdf

New notions are introduced in algebra in order to better study the congruences in number theory.

Author : W. B. Vasantha Kandasamy
Publisher : Infinite Study
Page : 455 pages
File Size : 47,7 Mb
Release : 2003
Category : Mathematics
ISBN : 9781931233743

Get Book

Smarandache Fuzzy Algebra by W. B. Vasantha Kandasamy Pdf

The author studies the Smarandache Fuzzy Algebra, which, like its predecessor Fuzzy Algebra, arose from the need to define structures that were more compatible with the real world where the grey areas mattered, not only black or white.In any human field, a Smarandache n-structure on a set S means a weak structure {w(0)} on S such that there exists a chain of proper subsets P(n-1) in P(n-2) in?in P(2) in P(1) in S whose corresponding structures verify the chain {w(n-1)} includes {w(n-2)} includes? includes {w(2)} includes {w(1)} includes {w(0)}, where 'includes' signifies 'strictly stronger' (i.e., structure satisfying more axioms).This book is referring to a Smarandache 2-algebraic structure (two levels only of structures in algebra) on a set S, i.e. a weak structure {w(0)} on S such that there exists a proper subset P of S, which is embedded with a stronger structure {w(1)}. Properties of Smarandache fuzzy semigroups, groupoids, loops, bigroupoids, biloops, non-associative rings, birings, vector spaces, semirings, semivector spaces, non-associative semirings, bisemirings, near-rings, non-associative near-ring, and binear-rings are presented in the second part of this book together with examples, solved and unsolved problems, and theorems. Also, applications of Smarandache groupoids, near-rings, and semirings in automaton theory, in error correcting codes, and in the construction of S-sub-biautomaton can be found in the last chapter.

Author : Florentin Smarandache
Publisher : Infinite Study
Page : 232 pages
File Size : 51,9 Mb
Release : 2016-11-12
Category : Mathematics
ISBN : 9781599734682

Get Book

The Encyclopedia of Neutrosophic Researchers, Vol. I by Florentin Smarandache Pdf

This is the first volume of the Encyclopedia of Neutrosophic Researchers, edited from materials offered by the authors who responded to the editor’s invitation. The 78 authors are listed alphabetically. The introduction contains a short history of neutrosophics, together with links to the main papers and books. Neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus and so on are gaining significant attention in solving many real life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistent, and indeterminacy. In the past years the fields of neutrosophics have been extended and applied in various fields, such as: artificial intelligence, data mining, soft computing, decision making in incomplete / indeterminate / inconsistent information systems, image processing, computational modelling, robotics, medical diagnosis, biomedical engineering, investment problems, economic forecasting, social science, humanistic and practical achievements.

Author : W. B. Vasantha Kandasamy
Publisher : American Research Press
Page : 203 pages
File Size : 45,9 Mb
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1461930006

Get Book

Smarandache Neutrosophic Algebraic Structures by W. B. Vasantha Kandasamy Pdf

Author : S. Suryoto,Harjito,T. Udjiani
Publisher : Infinite Study
Page : 7 pages
File Size : 52,7 Mb
Release : 2024-05-23
Category : Mathematics
ISBN : 8210379456XXX

Get Book

The algebraic structure on the neutrosophic triplet set by S. Suryoto,Harjito,T. Udjiani Pdf

The notion of the neutrosophic triplet was introduced by Smarandache and Ali. This notion is based on the fundamental law of neutrosophy that for an idea X, we have neutral of X denoted as neut(X) and anti of X denoted as anti(X). This paper studied a neutrosophic triplet set which is a collection of all triple of three elements that satisfy certain properties with some binary operation. Also given some interesting properties related to them. Further, in this paper investigated that from the neutrosophic triplet group can construct a classical group under multiplicative operation for ℤ𝑛 , for some specific n. These neutrosophic triplet groups are built using only modulo integer 2p, with p is an odd prime or Cayley table.

Author : Florentin Smarandache,Xiaohong Zhang,Mumtaz Ali
Publisher : Infinite Study
Page : 450 pages
File Size : 50,5 Mb
Release : 2024-05-23
Category : Mathematics
ISBN : 9783038974765

Get Book

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets, Volume II by Florentin Smarandache,Xiaohong Zhang,Mumtaz Ali Pdf

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity (i.e., element, concept, idea, theory, logical proposition, etc.), is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor . Based on neutrosophy, the neutrosophic triplets were founded; they have a similar form: (x, neut(x), anti(x), that satisfy some axioms, for each element x in a given set. This book contains the successful invited submissions to a special issue of Symmetry, reporting on state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets, and their algebraic structures—that have been defined recently in 2016, but have gained interest from world researchers, and several papers have been published in first rank international journals.

Author : W. B. Vasantha Kandasamy,Florentin Smarandache
Publisher : Infinite Study
Page : 128 pages
File Size : 40,6 Mb
Release : 2024-05-23
Category : Electronic
ISBN : 9781599732725

Get Book

Algebraic Structures on Fuzzy Unit Square and Neutrosophic Unit Square by W. B. Vasantha Kandasamy,Florentin Smarandache Pdf

In this book authors build algebraic structures on fuzzy unit semi-open square UF = {(a,b), with a, b in [0, 1)} and on neutrosophic unit semi-open square UN = {a+bI, with a, b in [0, 1)}. As distributive laws are not true, we are not in a position to develop several properties of rings, semirigs and linear algebras. Seven open conjectures are proposed.