scholarly journals A note on complex fuzzy subfield

2021 ◽  
Vol 21 (2) ◽  
pp. 1048
Author(s):  
Muhammad Gulzar ◽  
Fareeha Dilawar ◽  
Dilshad Alghazzawi ◽  
M. Haris Mateen
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Inverse Image

In this paper, we introduce idea of complex fuzzy subfield and discuss its various algebraic aspects. We prove that every complex fuzzy subfield generate two fuzzy fields and shows that intersection of two complex fuzzy subfields is also complex fuzzy subfields. We also present the concept of level subsets of complex fuzzy subfield and shows that level subset of complex fuzzy subfield form subfield.  Furthermore, we extend this idea to define the notion of the direct product of two complex fuzzy subfields and also investigate the homomorphic image and inverse image of complex fuzzy subfield.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

Additive Maps on Units of Rings

2018 ◽  
Vol 61 (1) ◽  
pp. 130-141
Author(s):  
Tamer Košan ◽  
Serap Sahinkaya ◽  
Yiqiang Zhou
Keyword(s):  
Recent Result ◽  
Direct Product ◽  
Homomorphic Image ◽  
Semilocal Ring ◽  
Additive Map ◽  
Exchange Rings ◽  
Additive Maps

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

scholarly journals A homomorphism theorem for projective planes

1971 ◽  
Vol 4 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Don Row
Keyword(s):  
Projective Plane ◽  
Homomorphic Image ◽  
Inverse Image ◽  
Projective Planes ◽  
Homomorphism Theorem ◽  
Central Collineations

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

scholarly journals Product structure in topological algebras

1975 ◽  
Vol 20 (4) ◽  
pp. 385-393
Author(s):  
Desmond A. Robbie
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Product Structure ◽  
Topological Algebras ◽  
Free Semigroups

It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.

scholarly journals Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 567 ◽  
Author(s):  
Hashem Bordbar ◽  
Young Bae Jun ◽  
Seok-Zun Song
Keyword(s):  
Homomorphic Image ◽  
Inverse Image ◽  
Closure Operation ◽  
Weak Closure ◽  
Closure Operations ◽  
Epimorphic Image

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

scholarly journals On 𝓠-regular semigroups

2018 ◽  
Vol 16 (1) ◽  
pp. 522-530
Author(s):  
Xinyang Feng
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Regular Semigroups ◽  
Subdirect Products ◽  
Maximum Group

AbstractIn this paper, we give some characterizations of 𝓠-regular semigroups and show that the class of 𝓠-regular semigroups is closed under the direct product and homomorphic images. Furthermore, we characterize the 𝓠-subdirect products of this class of semigroups and study the E-unitary 𝓠-regular covers for 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group. As an application of these results, we claim that the similar results on V-regular semigroups also hold.

scholarly journals Weak Essential Fuzzy Submodules Of Fuzzy Modules

2020 ◽  
Vol 33 (4) ◽  
pp. 65
Author(s):  
Hassan K. Marhon ◽  
Hatam Y. Khalaf
Keyword(s):  
Homomorphic Image ◽  
Inverse Image

        Throughout this paper, we introduce the notion of weak essential F-submodules of F-modules as a generalization of  weak essential submodules. Also we study the homomorphic image and inverse image of weak essential F-submodules.

scholarly journals Subdirect products of E–inversive semigroups

1990 ◽  
Vol 48 (1) ◽  
pp. 66-78 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Explicit Form ◽  
Direct Product ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Subdirect Products ◽  
Special Case ◽  
Maximum Group

AbstractA semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

scholarly journals CUBIC Z-SUBALGEBRAS IN Z-ALGEBRAS

2021 ◽  
Vol 2070 (1) ◽  
pp. 012085
Author(s):  
S. Sowmiya ◽  
P. Jeyalakshmi
Keyword(s):  
Homomorphic Image ◽  
Cartesian Product ◽  
Inverse Image

Abstract In this article, the notions of Cubic Z-Subalgebras in Z-algebras is introduced and some of their properties are investigated. The Z-homomorphic image and inverse image of cubic Z-Subalgebras in Z-algebras is investigated. Also, the cartesian product of cubic Z-Subalgebras in Z-algebras is also discussed.

Perfect extensions and derived algebras

1995 ◽  
Vol 60 (3) ◽  
pp. 775-796 ◽  
Author(s):  
Hajnal Andréka ◽  
Steven Givant ◽  
István Németi
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Homomorphic Image ◽  
Algebraic Logic ◽  
Negative Answer ◽  
Direct Products ◽  
Cylindric Algebra ◽  
Relation Algebras ◽  
Multiplicative Operator ◽  
Derived Algebras

Jónsson and Tarski [1951] introduced the notion of a Boolean algebra with (additive) operators (for short, a Bo). They showed that every Bo can be extended to a complete and atomic Bo satisfying certain additional conditions, and that any two complete, atomic extensions of satisfying these conditions are isomorphic over . Henkin [1970] extended these results to Boolean algebras with generalized (i.e., weakly additive) operators. The particular complete, atomic extension of studied by Jónsson and Tarski is called the perfect extension of , and is denoted by +. It is very useful in algebraic investigations of classes of algebras that are associated with logics.Interesting examples of Bos abound in algebraic logic, and include relation algebras, cylindric algebras, and polyadic and quasi-polyadic algebras (with or without equality). Moreover, there are several important constructions that, when applied to certain Bos, lead to other, derived Bos. Obvious examples include the formation of subalgebras, homomorphic images, relativizations, and direct products. Other examples include the Boolean algebra of ideal elements of a Bo, the neat β;-reduct of an α-dimensional cylindric algebra (β; < α), and the relation algebraic reduct of a cylindric algebra (of dimension at least 3). It is natural to ask about the relationship between the perfect extension of a Bo and the perfect extension of one of its derived algebras ′: Is the perfect extension of the derived algebra just the derived algebra of the perfect extension? In symbols, is (′)+ = (+)′? For example, is the perfect extension of a subalgebra, homomorphic image, relativization, or direct product, just the corresponding subalgebra, homomorphic image, relativization, or direct product of the perfect extension (up to isomorphisms)? Is the perfect extension of the Boolean algebra of ideal elements, or the neat reduct of a cylindric algebra, or the relation algebraic reduct of a cylindric algebra just the Boolean algebra of ideal elements, or the neat β;-reduct, or the relation algebraic reduct, of the perfect extension? We shall prove a general result in this direction; namely, if the derived algebra is constructed as the range of a relatively multiplicative operator, then the answer to our question is “yes”. We shall also give examples to show that in “infinitary” constructions, our question can have a spectacularly negative answer.

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