LingvoCore.com

homomorphism: meaning, definition, pronunciation and examples

Very Low
UK/ˌhɒm.ə(ʊ)ˈmɔː.fɪ.zəm/US/ˌhoʊ.moʊˈmɔːr.fɪ.zəm/

Technical/Formal

My Flashcards

Quick answer

What does “homomorphism” mean?

A structure-preserving map between two algebraic structures of the same type (e.

Audio

Pronunciation

Definition

Meaning and Definition

A structure-preserving map between two algebraic structures of the same type (e.g., groups, rings).

More broadly, any mapping between two sets that preserves the operations or relationships defined on them. In computer science, it can refer to a structure-preserving function between two data types.

Dialectal Variation

British vs American Usage

Differences

No significant differences in meaning or usage. Spelling follows respective norms (e.g., 'homomorphisms' vs. 'homomorphisms', no change).

Connotations

Purely technical term in both varieties.

Frequency

Equally rare outside of mathematics, computer science, and related academic fields.

Grammar

How to Use “homomorphism” in a Sentence

homomorphism from [structure A] to [structure B]homomorphism between [two structures]homomorphism of [structure type]

Vocabulary

Collocations

strong
group homomorphismring homomorphismnatural homomorphismdefine a homomorphismkernel of a homomorphism
medium
surjective homomorphisminjective homomorphismunique homomorphismexistence of a homomorphism
weak
continuous homomorphismalgebra homomorphismlinear homomorphismis a homomorphism

Examples

Examples of “homomorphism” in a Sentence

verb

British English

  • The function homomorphically maps the group elements.
  • We need to homomorphise these structures (very rare).

American English

  • The function homomorphically sends one ring to the other.
  • Can we homomorphize these algebras? (very rare).

adverb

British English

  • The two operations interact homomorphically.
  • The data was transformed homomorphically.

American English

  • The function acts homomorphically on the generated set.
  • The system processes signals homomorphically.

adjective

British English

  • The homomorphic property is essential for the proof.
  • We studied homomorphic encryption schemes.

American English

  • This mapping is homomorphic with respect to addition.
  • Homomorphic images of modules retain certain properties.

Usage

Meaning in Context

Business

Virtually never used.

Academic

Core term in abstract algebra, category theory, and universal algebra. Used in advanced mathematics and theoretical computer science papers.

Everyday

Not used in everyday conversation.

Technical

Used precisely in mathematics, cryptography (homomorphic encryption), and formal methods in software engineering.

Vocabulary

Synonyms of “homomorphism”

Strong

morphism (in category theory)

Neutral

structure-preserving mapmorphism

Weak

compatible function

Vocabulary

Antonyms of “homomorphism”

structure-breaking maparbitrary function (in this context)

Watch out

Common Mistakes When Using “homomorphism”

  • Misspelling as 'homomorphisim' or 'homomorfism'.
  • Confusing with 'homeomorphism' (a continuous bijection with a continuous inverse in topology).
  • Using it to mean any simple similarity in non-technical contexts.

FAQ

Frequently Asked Questions

An isomorphism is a bijective (one-to-one and onto) homomorphism. All isomorphisms are homomorphisms, but not all homomorphisms are isomorphisms. Isomorphisms indicate that the two structures are essentially identical.

Primarily yes, especially abstract algebra. However, the concept has been adopted in theoretical computer science (e.g., in type theory, formal languages) and notably in cryptography ('homomorphic encryption').

In group or ring theory, the kernel of a homomorphism is the set of elements that map to the identity element (or zero) in the target structure. It measures how much the homomorphism fails to be injective (one-to-one).

Yes. The function f from the integers (ℤ) under addition to the set {1, -1} under multiplication, defined by f(n) = 1 if n is even and -1 if n is odd, is a group homomorphism. f(a+b) = f(a) * f(b).

Homomorphism is usually technical/formal in register.

Homomorphism: in British English it is pronounced /ˌhɒm.ə(ʊ)ˈmɔː.fɪ.zəm/, and in American English it is pronounced /ˌhoʊ.moʊˈmɔːr.fɪ.zəm/. Tap the audio buttons above to hear it.

Learning

Memory Aids

Mnemonic

Think 'HOMO' (same) + 'MORPH' (shape/form) + 'ISM' (process). A homomorphism sends things to other things while keeping their 'shape' or structure the same.

Conceptual Metaphor

A TRANSLATION DICTIONARY between two mathematical languages that preserves grammatical rules (operations).

Practice

Quiz

Fill in the gap
A key property of a group is that f(a • b) = f(a) • f(b) for all elements a and b.
Multiple Choice

In which field is the term 'homomorphism' most precisely and fundamentally defined?