scholarly journals The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 411
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi ◽  
Khizar Hayat ◽  
Xiao-Peng Yang
Keyword(s):  
Group Theory ◽  
Continuous Functions ◽  
Vital Role ◽  
Ring Theory ◽  
Mathematical Tool ◽  
Algebraic Structures ◽  
Ring Homomorphisms ◽  
Extremum Problems ◽  
Convexity Theory ◽  
Fundamental Theorems

In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

From Equations to Structures: Linking Abstract Algebra and High-School Algebra for Secondary School Teachers

2019 ◽  
pp. 517-522
Author(s):  
Josephine Shamash
Keyword(s):  
Field Theory ◽  
High School ◽  
Group Theory ◽  
Secondary School ◽  
Secondary School Teachers ◽  
Abstract Algebra ◽  
School Curriculum ◽  
Algebraic Structures ◽  
High School Curriculum ◽  
Fundamental Theorems

The high-school curriculum in algebra deals mainly with the solution of different types of equations. Modern algebra has a completely different viewpoint and is concerned with algebraic structures and operations. The course Algebra: From Equations to Structures is part of an M.Sc. programme for Israeli secondary school mathematics teachers. It provides an introduction to algebraic structures and modern abstract algebra, and links abstract algebra to the high-school curriculum in algebra. It follows the historical attempts of mathematicians to solve polynomial equations of higher degrees, attempts which resulted in the development of group theory and field theory by Galois and Abel. This approach leads naturally to examining topics and fundamental theorems in both group theory and field theory. Along the historical “journey”, many other major results in algebra in the past 150 years are introduced, and current research in algebra is highlighted. We examine the relevance of the course to the teachers' work.

Image development in the framework of ξ –complex fuzzy morphisms

2021 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Unit Disk ◽  
Homomorphic Image ◽  
Significant Role ◽  
Original Form ◽  
Mathematical Tool ◽  
Fuzzy Subgroups ◽  
Fundamental Theorems

The study of complex fuzzy sets defined over the meet operator (ξ –CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ –CFS and propose the notion of complex fuzzy subgroups defined over ξ –CFS (ξ –CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –complex fuzzy subgroups and establish fundamental theorems of ξ –complex fuzzy morphisms. In addition, we effectively apply the idea of ξ –complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ –complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

scholarly journals Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 321 ◽  
Author(s):  
Mehmet Çelik ◽  
Moges Shalla ◽  
Necati Olgun
Keyword(s):  
Group Theory ◽  
Significant Role ◽  
Classical Group ◽  
Algebraic Structures ◽  
Algebraic Systems ◽  
Homomorphism Theorem ◽  
Special Cases ◽  
Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

scholarly journals Green's relations for regular elements of sandwich semigroups, II; semigroups of continuous functions

1978 ◽  
Vol 25 (1) ◽  
pp. 45-65 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Continuous Functions ◽  
Maximal Subgroups ◽  
Green’S Relations ◽  
Boolean Ring ◽  
Regular Elements ◽  
Ring Homomorphisms ◽  
Special Cases ◽  
Green's Relations ◽  
Sandwich Semigroup ◽  
Sandwich Semigroups

AbstractA sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

Note on the Singular Submodule

1970 ◽  
Vol 13 (4) ◽  
pp. 441-442
Author(s):  
D. Fieldhouse
Keyword(s):  
Ring Theory ◽  
New Technique ◽  
Ring Homomorphisms ◽  
A New Technique ◽  
Singular Ideal

One very interesting and important problem in ring theory is the determination of the position of the singular ideal of a ring with respect to the various radicals (Jacobson, prime, Wedderburn, etc.) of the ring. A summary of the known results can be found in Faith [3, p. 47 ff.] and Lambek [5, p. 102 ff.]. Here we use a new technique to obtain extensions of these results as well as some new ones.Throughout we adopt the Bourbaki [2] conventions for rings and modules: all rings have 1, all modules are unital, and all ring homomorphisms preserve the 1.

scholarly journals Paul Moritz Cohn. 8 January 1924 — 20 April 2006

2014 ◽  
Vol 60 ◽  
pp. 127-150 ◽  
Author(s):  
George Bergman ◽  
Trevor Stuart
Keyword(s):  
Group Theory ◽  
Trinity College ◽  
London Mathematical Society ◽  
Ring Theory ◽  
Concentration Camps ◽  
Entrance Examination ◽  
English School ◽  
Lie Rings ◽  
Chicken Farm ◽  
Work First

Paul Cohn was born in Hamburg, where he lived until he was 15 years of age. However, in 1939, after the rise of the Nazis and the growing persecution of the Jews, his parents, James and Julia Cohn, sent him to England by Kindertransport. They remained behind and Paul never saw them again; they perished in concentration camps. In England, being only 15 years old, he was directed to work first on a chicken farm but later as a fitter in a London factory. His academic talents became clear and he was encouraged by the refugee committee in Dorking and by others to continue his education by studying for the English School Certificate Examinations to sit the Cambridge Entrance Examination. He was awarded an Exhibition to study mathematics at Trinity College. After receiving his PhD in 1951, Paul Cohn went from strength to strength in algebra and not only became a world leader in non-commutative ring theory but also made important contributions to group theory, Lie rings and semigroups. He was much admired, and he travelled widely to collaborate with other algebraists. Moreover, he was a great supporter of the London Mathematical Society, serving as its President from 1982 to 1984.

scholarly journals Applications of Soft Union Sets in the Ring Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yongwei Yang ◽  
Xiaolong Xin ◽  
Pengfei He
Keyword(s):  
Ring Theory ◽  
Algebraic Structures ◽  
Soft Sets ◽  
Quotient Rings ◽  
Isomorphism Theorems ◽  
Algebraic Tool

The aim of the paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion ofλ,μ-soft union rings which is a generalization of that of soft union rings is proposed. By introducing the notion of soft cosets, soft quotient rings based onλ,μ-soft union ideals are established. Moreover, through discussing quotient soft subsets, an approach for constructing quotient soft union rings is made. Finally, isomorphism theorems ofλ,μ-soft union rings related to invariant soft sets are discussed.

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