scholarly journals Simultaneous Extension of Partial Endomorphisms of Groups

1954 ◽  
Vol 2 (1) ◽  
pp. 37-46 ◽  
Author(s):  
C. G. Chehata
Keyword(s):  
Simultaneous Extension ◽  
Homomorphic Mapping

Let μ be a homomorphic mapping of some subgroup A of the group G onto a subgroup Ḃ (not necessarily distinct from A) of G; then we call μ a partial endomorphism of G. If A coincides with G, that is, if the homomorphism is defined on the whole of G, we speak of a total endomorphism; this is what is usually called an endomorphism of G. A partial (or total) endomorphism μ*extends or continues a partial endomorphism μ if the domain of μ* contains the domain of μ, that is, μ* is defined for (at least) all those elements for which μ. is defined, and moreover μ* coincides with μ where μ is defined.

On Commutative Continuation of Partial Endomorphisms of Groups

1965 ◽  
Vol 17 ◽  
pp. 429-433 ◽  
Author(s):  
C. G. Chehata
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Homomorphic Mapping ◽  
Necessary And Sufficient

Given a homomorphic mapping θ of a subgroup A of a group G onto another subgroup B of G, necessary and sufficient conditions for the existence of a supergroup G* of G and an endomorphism θ* of G* such that θ* coincides with θ on A were derived by B. H. Neumann and Hanna Neumann (3). The homomorphism θ is called a partial endomorphism of G and θ* is said to continue, or extend, θ. Necessary and sufficient conditions for the simultaneous continuation of two partial endomorphisms of a group G to total endomorphisms of one supergroup G* ⊇ G were derived by the author (2).

A Continuous Extension Operator for Convex Metrics

2010 ◽  
Vol 53 (4) ◽  
pp. 719-729
Author(s):  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Extension Operator ◽  
Uniform Topology ◽  
Simultaneous Extension ◽  
Peano Continuum ◽  
Convex Metrics

AbstractWe consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices

1952 ◽  
Vol 4 ◽  
pp. 31-42 ◽  
Author(s):  
J. K. Goldhaber
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Characteristic Roots ◽  
Finite Set ◽  
Homomorphic Mapping ◽  
Necessary And Sufficient

The problem of determining the conditions under which a finite set of matrices A1A2, … , Ak has the property that their characteristic roots λ1j, λ2j, … , λki (j = 1, 2, …, n) may be so ordered that every polynomial f(A1A2 … , Ak) in these matrices has characteristic roots f(λ1j, λ2j …,λki) (j = 1, 2, … , n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (Ai〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.

scholarly journals Research on the Properties of Lattice-Valued Finite State Machines in Computer Technology

2021 ◽  
Vol 2066 (1) ◽  
pp. 012063
Author(s):  
Zhonggang Hu
Keyword(s):  
Finite State Machine ◽  
Finite State Machines ◽  
State Machine ◽  
Exchange Capacity ◽  
State Machines ◽  
Strong Connectivity ◽  
Algebraic Properties ◽  
Finite State ◽  
Strongly Connected ◽  
Homomorphic Mapping

Abstract In this paper, we discuss some algebraic properties of Lattice valued finite state machine and prove that if there are homomorphic mapping satisfying certain conditions between two Lattice valued finite state machines, the first one is strongly connected (cycle), then then the second one is the same. And if the homomorphism is strongly homomorphic, one of the Lattice valued finite state machines is complete if and only if another Lattice valued finite state machine is complete. Discuss the completeness, strong connectivity, circulation and exchange capacity between the product of a Lattice valued finite state machine and the original Lattice valued finite state machine and get some results.

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