Generally the study of algebraic 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.
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 …
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 occurin our everyday'slife, that's why we study them in this book. Thus, as a particular case: A Smarandache groupoid is a groupoid G which has a proper subset S contained in G such that S under the operation of G is a semigroup.
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 life, that's why we study them in this book. Thus, as a particular case: A Smarandache semigroup is a semigroup A which has a proper subset B contained in A that is a group (with respect to the same binary operation on A).
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 which is embedded with a stronger structure S. By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our every day's life, that's why we study them in this book. Thus, as three particular cases: 1) A Smarandache semiring is a semiring A such that a proper subset B of A is a semifield (with respect to the same induced operation). 2) A Smarandache semifiel…
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 which is embedded with a stronger structure S. By a proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our every day's life, that's why we study them in this book. Thus, as a particular case, we investigate the theory of linear algebra and Smarandache linear algebra.
When implementing the DSmT, a difficulty may arise from the possible huge dimension of hyperpower sets, which are indeed free structures. However, it is possible to reduce the dimension of these structures by involving logical constraints. In this chapter, the logical constraints will be related to a predefined order over the logical propositions. The use of such orders and their resulting logical constraints will ensure a great reduction of the model complexity. Such results will be applied to the definition of continuous DSm models. In particular, a simplified description of the continuous i…
This book has seven chapters. In Chapter one, an elaborate recollection of Smarandache structures like S-semigroups, S-loops, and S-groupoids is given. It also gives notions about N-ary algebraic stuctures and their Smarandache analogue, Neutrosophic structures viz. groups, semigroups, groupoids and loops are given in Chapter one to make the book a self-contained one. For the first time, S-neutrosophic groups and S-neutrosophic N-groups are introduced in Chapter two and their properties are given. S-neutrosophic semigroups and S-neutrosophic N-semigroups are defined and discussed in Chapter th…
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This is the third part on multi-spaces concertrating on Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries. In where, the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries.
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This paper is the first part on characterizing multi-spaces. Various algebraic multi-spaces with structures such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems are discussed and new results are obtained.
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 which is embedded with a stronger structure S. By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our everyday's life thats why we study them in this book. Thus, as two particular cases: A Smarandache Ring of level I (S-ring I) is a ring R that contains a proper subset that is a field with respect to the operations induced. A Smarandache Ring of level…
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This is second part on multi-spaces. Many conceptions in graphs are generalized by Smarandache's notion, such as multi-voltage graphs, Cayley graphs of a finite multi-group,multi-embedding of a graph in an $n$-manifold, graph phase, $...$, etc.. New results are obtained.
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 non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c belonging to R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b. A Smarandache …
A Smarandache multi-space is a union of $n$ different spaces equipped with some different structures for an integer $n\geq 2$, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This is the 4th part of multi-spaces considering applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for $p$-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces.
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 …
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 which is embedded with a stronger structure S. By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our everyday's life, thats why we study them in this book. As an example: A non-empty set L is said to form a loop, if on L is defined a binary operation called product, denoted by '.' such that: for all a, b belonging to L we have a . b belonging to L (c…
A Smarandache multi-space is a union of $n$ spaces $A_1,A_2,..., A_n$ with some additional conditions holding. Combining classical of a group with Smarandache multi-spaces, the conception of a multi-group space is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, filed, body, $...$, etc.. Similar to groups, some characteristics of a multi-group space are obtained in this paper.
The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005)we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bial…
