Isomorphism and Weak Isomorphism

AbstractIn the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either irivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C(T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.

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Investigating the Short-Circuit Problem Using the Planarity Index of Complex q-Rung Orthopair Fuzzy Planar Graphs

Complexity ◽

10.1155/2021/8295997 ◽

2021 ◽

Vol 2021 ◽

pp. 1-22

Author(s):

Abrar Hussain ◽

Ahmed Alsanad ◽

Kifayat Ullah ◽

Zeeshan Ali ◽

Muhammad Kamran Jamil ◽

...

Keyword(s):

Planar Graphs ◽

Short Circuit ◽

Dual Graph ◽

Practical Applications ◽

Weak Isomorphism ◽

Uncertain Knowledge ◽

Effective Role ◽

Short Circuits ◽

Set Up ◽

Planarity Index

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted.

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On Weak Isomorphism of Rooted Vertex-Colored Graphs

Graph-Theoretic Concepts in Computer Science - Lecture Notes in Computer Science ◽

10.1007/978-3-030-00256-5_22 ◽

2018 ◽

pp. 266-278

Author(s):

Lars Jaffke ◽

Mateus de Oliveira Oliveira

Keyword(s):

Colored Graphs ◽

Weak Isomorphism

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Enumeration of weak isomorphism classes of signed graphs

Journal of Graph Theory ◽

10.1002/jgt.3190040202 ◽

1980 ◽

Vol 4 (2) ◽

pp. 127-144 ◽

Cited By ~ 15

Author(s):

Tadeusz Sozański

Keyword(s):

Signed Graphs ◽

Isomorphism Classes ◽

Weak Isomorphism

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Extending abelian groups to rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700036132 ◽

2007 ◽

Vol 82 (3) ◽

pp. 297-314 ◽

Cited By ~ 2

Author(s):

Lynn M. Batten ◽

Robert S. Coulter ◽

Marie Henderson

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Equivalence Relation ◽

Binary Operation ◽

Sufficient Conditions ◽

Additive Group ◽

Necessary And Sufficient Conditions ◽

Odd Order ◽

Weak Isomorphism ◽

Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

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