INTRODUCTION TO CLASSICAL ALGEBRA

M-Hazy Vector Spaces over M-Hazy Field

Mathematics ◽

10.3390/math9101118 ◽

2021 ◽

Vol 9 (10) ◽

pp. 1118

Author(s):

Faisal Mehmood ◽

Fu-Gui Shi

Keyword(s):

Vector Space ◽

Linear Transformation ◽

Binary Operation ◽

Vector Spaces ◽

Fundamental Properties ◽

Classical Algebra ◽

Important Development ◽

Fuzzy Algebra ◽

Convex Spaces

The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called M-hazy vector spaces over M-hazy field. Some fundamental properties of M-hazy field, M-hazy vector spaces, and M-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of M-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that M-fuzzifying convex spaces are induced by an M-hazy subspace of M-hazy vector space.

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Algebra of Nonlocal Charges in Supersymmetric Nonlinear Sigma Models

International Journal of Modern Physics A ◽

10.1142/s0217751x97000487 ◽

1997 ◽

Vol 12 (02) ◽

pp. 419-436 ◽

Cited By ~ 2

Author(s):

L. E. Saltini ◽

A. Zadra

Keyword(s):

Sigma Models ◽

Sigma Model ◽

Graphic Method ◽

Two Dimensional ◽

Conserved Charges ◽

Classical Algebra ◽

Dirac Brackets ◽

Yangian Algebra ◽

Graphic Methods ◽

Nonlinear Sigma Models

We propose a graphic method to derive the classical algebra (Dirac brackets) of nonlocal conserved charges in the two-dimensional supersymmetric nonlinear O(N) sigma model. As in the purely bosonic theory we find a cubic Yangian algebra. We also consider the extension of graphic methods to other integrable theories.

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On the classical algebra

Journal of Physics A General Physics ◽

10.1088/0305-4470/29/7/016 ◽

1996 ◽

Vol 29 (7) ◽

pp. 1453-1463

Author(s):

L Chao ◽

Q P Liu

Keyword(s):

Classical Algebra

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Classical algebra

Moonshine beyond the Monster ◽

10.1017/cbo9780511535116.002 ◽

2009 ◽

pp. 14-103

Author(s):

Terry Gannon

Keyword(s):

Classical Algebra

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REDUCING FUZZY ALGEBRA TO CLASSICAL ALGEBRA

New Mathematics and Natural Computation ◽

10.1142/s1793005705000020 ◽

2005 ◽

Vol 01 (01) ◽

pp. 27-64 ◽

Cited By ~ 6

Author(s):

ARTHUR WEINBERGER

Keyword(s):

Modular Lattice ◽

Unit Interval ◽

Embedding Theorem ◽

Surprising Result ◽

Normal Subgroups ◽

Classical Algebra ◽

Fuzzy Algebra ◽

Related Algebra ◽

Main Ideas ◽

Fuzzy Subsets

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras. The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets. The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀. The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

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