Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

Yuan, Wangtao; Zhang, Xiao­hong

doi:10.3390/math8020204

Cited by 4 publications

(6 citation statements)

References 19 publications

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“…Table 12, then (S, *) is a variant CA-groupoid and 1 is a quasi-right unit element in S. Obviously, S isn't commutative. Looking at the above example carefully, we find that: (1) the element 1 as a quasi-right unit element of S, does not appear in the operation table; (2) in the operation table, the first row is the same as the third row; (3) if we change the first row of the operation table to {1, 2, 3, 4, 5}, we will get a commutative semigroup (S, +) (as shown in Table 13). These are all interesting phenomena.…”

Section: Variant Ca-groupoidsmentioning

confidence: 99%

“…(2) Suppose that S is constructed according to the method described in (2), then for all x, y, z∈S 1 , x*yz = z*xy = y*zx, and:…”

Section: Proposition 7 (1) If S Is a Variant Ca-groupoid Then S Isnmentioning

confidence: 99%

“…An algebraic structure is called a groupoid, if it is well-defined regarding an operation on it. A groupoid satisfying the "cyclic associative law" (that is, x(yz) = y(zx)) is called a cyclic associative groupoid, or simply CA-groupoid [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Later, Kleinfeld [7] and Behn [8,9] studied the rings satisfying the cyclic associative law, and Iqbal et al [10] studied the AG-groupoids satisfying the cyclic associative law. It is on the basis of these researches that we start to systematically study the groupoids satisfying the cyclic binding law (CA-groupoids) in [1,2], in order to provide a common basis for the research of related algebraic systems.…”

Section: Introductionmentioning

confidence: 99%

“…As a continuation of [1,2], this paper focuses on various cancellation properties of CA-groupoids and a special class of CA-groupoids. In many algebraic systems (such as semigroups, commutative semigroups and AG-groupoids), the cancellation, quasi-cancellation and power cancellation properties have important research value (see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]).…”

Section: Introductionmentioning

confidence: 99%

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Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids

Zhang

Smarandache

2020

Symmetry

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Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established.

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Section: Variant Ca-groupoidsmentioning

confidence: 99%

“…(2) Suppose that S is constructed according to the method described in (2), then for all x, y, z∈S 1 , x*yz = z*xy = y*zx, and:…”

Section: Proposition 7 (1) If S Is a Variant Ca-groupoid Then S Isnmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids

Zhang

Smarandache

2020

Symmetry

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Study of Two Kinds of Quasi AG-Neutrosophic Extended Triplet Loops

Chen

2021

Journal of Mathematics

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Abel-Grassmann’s groupoid and neutrosophic extended triplet loop are two important algebraic structures that describe two kinds of generalized symmetries. In this paper, we investigate quasi AG-neutrosophic extended triplet loop, which is a fusion structure of the two kinds of algebraic structures mentioned above. We propose new notions of AG-(l,r)-Loop and AG-(r,l)-Loop, deeply study their basic properties and structural characteristics, and prove strictly the following statements: (1) each strong AG-(l,r)-Loop can be represented as the union of its disjoint sub-AG-groups, (2) the concepts of strong AG-(l,r)-Loop, strong AG-(l,l)-Loop, and AG-(l,lr)-Loop are equivalent, and (3) the concepts of strong AG-(r,l)-Loop and strong AG-(r,r)-Loop are equivalent.

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On type-2 cyclic associative groupoids and inflationary pseudo general residuated lattices

An,

Chen

2024

IFS

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This paper explores the relationship between fuzzy logic algebra and non associative groupoid. As a groupoid which can satisfy type-2 cyclic associative (T2CA) law, T2CA-groupoid is characterized by generalized symmetry. Fuzzy logic algebra is a major direction in the study of fuzzy logic. Residuated lattices are a class of fuzzy logic algebras with widespread applications. The inflationary pseudo general residuated lattice (IPGRL), a generalization of the residuated lattice, does not need to satisfy the associative law and commutative law. Moreover, the greatest element of IPGRL is no longer the identity element. In this paper, the notion of T2CA-IPGRL (IPGRL in T2CA-groupoid) is proposed and its properties are investigated in combination with the study of IPGRL and T2CA-groupoid. In addition, the generalized symmetry and regularity of T2CA-groupoid are investigated based on the characteristics of commutative elements. Meanwhile, the decomposition of T2CA-root of band with T2CA-unipotent radical is studied as well. The result shows that every T2CA-root of band is the disjoint union of T2CA-unipotent radicals.

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Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

Cited by 4 publications

References 19 publications

Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids

Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids

Study of Two Kinds of Quasi AG-Neutrosophic Extended Triplet Loops

On type-2 cyclic associative groupoids and inflationary pseudo general residuated lattices

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