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# homomorphisms

## Definitions

• n. Plural form of homomorphism.

## Etymologies

Sorry, no etymologies found.

## Examples

• The basic maps between structures of the same signature K are called homomorphisms, defined as follows.

First-order Model Theory

• In the 1930s Garrett Birkhoff established the fundamental results of equational logic, namely (1) equational classes of algebras are precisely the classes closed under homomorphisms, subalgebras and direct products, and (2) equational logic is based on five rules: reflexivity, symmetry, transitivity, replacement, and substitution.

• Although there are maps between the respective sets of observables, Scheibe considers this as a case of incommensurability, since these maps are not Lie algebra homomorphisms, see Scheibe (1999, 174).

Structuralism in Physics

• Not only does he study systems of objects or whole classes of such systems; and not only does he attempt to identify basic concepts; Dedekind also tends to do both, often in conjunction, by considering mappings on the systems studied, especially structure-preserving mappings (homomorphisms etc.) and what is invariant under them.

Dedekind's Contributions to the Foundations of Mathematics

• However F also maps functions to homomorphisms, mapping f to its unique extension as a homomorphism, while U maps homomorphisms to functions, namely the homomorphism itself as a function.

Algebra

• The category AbGrp with objects abelian groups and morphisms group homomorphisms, i.e. (1, Ã—,?) homomorphisms

Category Theory

• The categories Lat and Bool with objects lattices and Boolean algebras, respectively, and morphisms structure preserving homomorphisms, i.e., (¤, ¥, §, ¨) homomorphisms.

Category Theory

• Similarly, natural transformations between models of a theory yield the usual homomorphisms of structures in the traditional set theoretical framework.

Category Theory

• The notation C (A, B) is generally used to denote the set of all homomorphisms from A to B.

Algebra

• Hence the homomorphisms between B and G compose in either order to identities, which makes them isomorphisms.

Algebra

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