Given two sets with some structure imposed on it - for instance, a relation, or an operation, or an algebra - a homomorphism is a mapping from one to the other that preserves that structure.

E.g., the natural numbers can be mapped to the even natural numbers by the mapping x -> 2x; this is a homomorphism with respect to the usual ordering of numbers.

x -> 2^x is a homomorphism to map addition onto multiplication.

Isomorphisms are the most common.

Ho`mo*mor"phism (?), n. [See Homomorphous.]

1. Biol.

Same as Homomorphy.

2. Bot.

The possession, in one species of plants, of only one kind of flowers; -- opposed to heteromorphism, dimorphism, and trimorphism.

3. Zool.

The possession of but one kind of larvae or young, as in most insects.

<-- 4. (Math) A special type of mapping of one mathematical set into or onto another set . . . -->

 

© Webster 1913.

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