Bilagebraic Structures And Smarandache Bialgebraic Structures

Author : W. B. Vasantha Kandasamy,Florentin Smarandache
Publisher : Infinite Study
Page : 209 pages
File Size : 50,6 Mb
Release : 2005-01-01
Category : Mathematics
ISBN : 9781931233057

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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
Publisher : Infinite Study
Page : 141 pages
File Size : 44,6 Mb
Release : 2009-01-01
Category : Mathematics
ISBN : 9781599730851

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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 : 199 pages
File Size : 46,8 Mb
Release : 2012
Category : Algebra, Boolean
ISBN : 9781599732169

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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 : W. B. Vasantha Kandasamy, Florentin Smarandache
Publisher : Infinite Study
Page : 168 pages
File Size : 40,5 Mb
Release : 2013
Category : Mathematics
ISBN : 9781599732121

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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 : 5 pages
File Size : 45,5 Mb
Release : 2024-06-23
Category : Electronic
ISBN : 8210379456XXX

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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,Florentin Smarandache
Publisher : Infinite Study
Page : 172 pages
File Size : 52,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9781599731353

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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 : 128 pages
File Size : 50,6 Mb
Release : 2024-06-23
Category : Electronic
ISBN : 9781599732725

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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.

Author : Florentin Smarandache
Publisher : Infinite Study
Page : 16 pages
File Size : 43,7 Mb
Release : 2024-06-23
Category : Mathematics
ISBN : 8210379456XXX

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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 : W. Charles Holland
Publisher : CRC Press
Page : 214 pages
File Size : 52,9 Mb
Release : 2001-04-01
Category : Mathematics
ISBN : 9056993259

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Ordered Algebraic Structures by W. Charles Holland Pdf

This book is an outcome of the conference on ordered algebraic structures held at Nanjing. It covers a range of topics: lattice theory, ordered semi groups, partially ordered groups, totally ordered groups, lattice-ordered groups, and ordered fields.

Author : W. B. Vasantha Kandasamy
Publisher : American Research Press
Page : 203 pages
File Size : 53,5 Mb
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1461930006

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

Author : Florentin Smarandache
Publisher : Infinite Study
Page : 13 pages
File Size : 42,5 Mb
Release : 2023-01-01
Category : Mathematics
ISBN : 8210379456XXX

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Introduction to the Symbolic Plithogenic Algebraic Structures (revisited) by Florentin Smarandache Pdf

In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.

Author : W. B. Vasantha Kandasamy
Publisher : Infinite Study
Page : 272 pages
File Size : 51,7 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 9781931233712

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Bilagebraic Structures and Smarandache Bialgebraic Structures by W. B. Vasantha Kandasamy Pdf

Generally the study of algebraic structures deals with the concepts like groups, semigroups, groupoids, loops, rings, near-rings, semirings, and vector spaces. The study of bialgebraic structures deals with the study of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binear-rings, bisemirings and bivector spaces. A complete study of these bialgebraic structures and their Smarandache analogues is carried out in this book. For examples: A set (S, +, *) with two binary operations ?+? and '*' is called a bisemigroup of type II if there exists two proper subsets S1 and S2 of S such that S = S1 U S2 and(S1, +) is a semigroup.(S2, *) is a semigroup. Let (S, +, *) be a bisemigroup. We call (S, +, *) a Smarandache bisemigroup (S-bisemigroup) if S has a proper subset P such that (P, +, *) is a bigroup under the operations of S. Let (L, +, *) be a non empty set with two binary operations. L is said to be a biloop if L has two nonempty finite proper subsets L1 and L2 of L such that L = L1 U L2 and(L1, +) is a loop, (L2, *) is a loop or a group. Let (L, +, *) be a biloop we call L a Smarandache biloop (S-biloop) if L has a proper subset P which is a bigroup. Let (G, +, *) be a non-empty set. We call G a bigroupoid if G = G1 U G2 and satisfies the following:(G1 , +) is a groupoid (i.e. the operation + is non-associative), (G2, *) is a semigroup. Let (G, +, *) be a non-empty set with G = G1 U G2, we call G a Smarandache bigroupoid (S-bigroupoid) if G1 and G2 are distinct proper subsets of G such that G = G1 U G2 (neither G1 nor G2 are included in each other), (G1, +) is a S-groupoid.(G2, *) is a S-semigroup.A nonempty set (R, +, *) with two binary operations ?+? and '*' is said to be a biring if R = R1 U R2 where R1 and R2 are proper subsets of R and (R1, +, *) is a ring, (R2, +, ?) is a ring.A Smarandache biring (S-biring) (R, +, *) is a non-empty set with two binary operations ?+? and '*' such that R = R1 U R2 where R1 and R2 are proper subsets of R and(R1, +, *) is a S-ring, (R2, +, *) is a S-ring.

Author : W. B. Vasantha Kandasamy
Publisher : Infinite Study
Page : 455 pages
File Size : 51,5 Mb
Release : 2003
Category : Mathematics
ISBN : 9781931233743

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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 : S. Suryoto,Harjito,T. Udjiani
Publisher : Infinite Study
Page : 7 pages
File Size : 49,5 Mb
Release : 2024-06-23
Category : Mathematics
ISBN : 8210379456XXX

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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 : Arsham Borumand Saeid
Publisher : Infinite Study
Page : 65 pages
File Size : 52,8 Mb
Release : 2024-06-23
Category : Electronic
ISBN : 9781599732411

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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.