scholarly journals VARIETIES WHOSE TOLERANCES ARE HOMOMORPHIC IMAGES OF THEIR CONGRUENCES

2012 ◽  
Vol 87 (2) ◽  
pp. 326-338 ◽  
Author(s):  
GÁBOR CZÉDLI ◽  
EMIL W. KISS
Keyword(s):  
Homomorphic Image ◽  
Tolerance Relation ◽  
Varieties Of Lattices ◽  
Majority Term

AbstractThe homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

scholarly journals L-fuzzy ideals and L-fuzzy subalgebras of Novikov algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1538-1546
Author(s):  
Xin Zhou ◽  
Liangyun Chen ◽  
Yuan Chang
Keyword(s):  
Fuzzy Sets ◽  
Homomorphic Image ◽  
Quotient Algebra ◽  
Algebraic Properties ◽  
Novikov Algebras ◽  
Fuzzy Ideal

Abstract In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L-fuzzy ideals and L-fuzzy subalgebras. We get a sufficient and neccessary condition such that an L-fuzzy subspace is an L-fuzzy ideal. Moreover, we show that the quotient algebra A/μ of the L-fuzzy ideal μ is isomorphic to the algebra A/Aμ of the non-fuzzy ideal Aμ. Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L-fuzzy ideal.

scholarly journals A note on central group extensions

1973 ◽  
Vol 15 (4) ◽  
pp. 428-429 ◽  
Author(s):  
G. J. Hauptfleisch
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Free Group ◽  
Central Extension ◽  
Dihedral Group ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

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 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.

scholarly journals Inverse semigroups and their natural order

1978 ◽  
Vol 19 (1) ◽  
pp. 59-65 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Homomorphic Image ◽  
Point Of View ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Theoretical Point ◽  
Trivial Group

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

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.

Semiregular Modules and Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 1105-1120 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Projective Module ◽  
Structure Theorem ◽  
Projective Cover ◽  
Finitely Generated

Mares [9] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [16] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

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