Homomorphic vs. Isomorphic — What's the Difference?
By Fiza Rafique & Maham Liaqat — Updated on May 9, 2024
Homomorphic refers to a mathematical or structural similarity that preserves some structure between two sets, while isomorphic implies a perfect structural equivalence, where two structures are identical in their form or properties.
Difference Between Homomorphic and Isomorphic
Table of Contents
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Key Differences
Homomorphic structures maintain a specific relationship between two sets by preserving certain properties or operations through a mapping. Isomorphic structures, on the other hand, have a one-to-one correspondence that ensures an exact structural match.
Homomorphic mappings (homomorphisms) retain essential operations like addition or multiplication between algebraic structures but don't necessarily map each element uniquely. Isomorphic mappings (isomorphisms) are bijective, ensuring every element corresponds perfectly to an element in the other set.
Homomorphic relationships are more general and may be found in algebra, topology, and computer science. Isomorphic relationships are stricter and signify full structural identity, making them particularly important in group theory and graph theory.
Homomorphisms allow partial similarities where some structure remains consistent. Isomorphisms require that all structural aspects match exactly, making the two structures indistinguishable in their form.
Homomorphic transformations are useful in simplifying problems by retaining only crucial aspects, while isomorphic transformations are used when exact replicas of structures are required.
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Comparison Chart
Definition
Structural similarity preserving some properties
Exact structural equivalence between two sets
Mapping Type
Not necessarily one-to-one
One-to-one, bijective
Structure
Partially similar
Perfectly identical
Mathematical Use
Algebraic structures, encryption
Group theory, graph theory
Application Scope
Broader, general structural similarity
Narrower, requiring strict structural identity
Compare with Definitions
Homomorphic
Refers to a mapping that preserves certain properties between two algebraic structures.
The mapping from integers to even integers is homomorphic for addition.
Isomorphic
Describes a mapping that shows a perfect, one-to-one correspondence between structures.
The two groups are isomorphic because they have identical multiplication tables.
Homomorphic
Retaining a consistent relationship between data structures.
The database transformation was homomorphic to maintain search functionality.
Isomorphic
Having an identical structural pattern in two systems.
The isomorphic graphs have the same number of vertices and edges.
Homomorphic
Describing a transformation that conserves core features.
The homomorphic function maintained the modular arithmetic structure.
Isomorphic
Indicating a bijective and structure-preserving relationship.
The two algebraic rings are isomorphic due to their similar properties.
Homomorphic
Denoting partial structural similarity.
The homomorphic encryption algorithm allows computations without decrypting data.
Isomorphic
Denoting structures that can be mapped precisely onto each other.
The molecular structures are isomorphic, meaning they share identical bonding arrangements.
Homomorphic
Maintaining a consistent pattern or operation in mathematical sets.
The polynomial function is homomorphic under composition.
Isomorphic
Showing full equivalence between two mathematical models.
The vector spaces are isomorphic as they have the same dimension and basis.
Homomorphic
(Mathematics) A transformation of one set into another that preserves in the second set the operations between the members of the first set.
Isomorphic
(Biology) Having a similar structure or appearance but being of different ancestry.
Homomorphic
(Biology) Similarity of external form or appearance but not of structure or origin.
Isomorphic
Related by an isomorphism.
Homomorphic
(Zoology) A resemblance in form between the immature and adult stages of an animal.
Isomorphic
(mathematics) Related by an isomorphism; having a structure-preserving one-to-one correspondence.
Homomorphic
(algebra) Of or pertaining to homomorphism; having a homomorphism.
Isomorphic
(biology) Having a similar structure or function to something that is not related genetically or through evolution.
Homomorphic
Characterized by homomorphism.
Isomorphic
Having identical relevant structure; being structure-preserving while undergoing certain invertible transformations.
Isomorphic
Isomorphous.
Isomorphic
Alike in form; exhibiting isomorphism.
Isomorphic
Of or pertaining to sets related by an isomorphism.
Isomorphic
Having similar appearance but genetically different
Common Curiosities
What are examples of homomorphic structures?
Examples include homomorphic encryption in computer science and homomorphisms between groups or rings in mathematics.
Do isomorphic structures imply similarity in function?
Yes, they imply perfect similarity in structure and function.
Are isomorphisms always bijective?
Yes, isomorphisms are always bijective, ensuring a one-to-one correspondence.
What properties do homomorphic mappings preserve?
They preserve operations like addition, multiplication, or other relevant functions.
Why are isomorphisms important in group theory?
They classify groups by revealing structural similarities that simplify analysis.
Can homomorphic structures become isomorphic?
Not necessarily. Homomorphic structures only preserve some aspects, while isomorphic structures require an exact match.
Why is homomorphism more common than isomorphism?
Homomorphic mappings are more flexible, as they preserve essential but not all properties, making them broadly applicable.
Are isomorphic structures identical in every sense?
They are identical in the structural sense required by their mathematical context.
Is an isomorphic structure unique?
Isomorphic structures are not unique individually but are unique as a pair due to their structural correspondence.
Can two graphs be homomorphic but not isomorphic?
Yes, a graph can have a homomorphic mapping to another without being structurally identical.
How does homomorphic encryption work?
It allows computations on encrypted data while preserving the encrypted result's relationship to the plaintext data.
Are homomorphic mappings reversible?
Not always, as they are not necessarily one-to-one.
Can homomorphic mappings simplify data processing?
Yes, homomorphic mappings help by retaining crucial information while reducing complexity.
What areas of mathematics commonly use isomorphisms?
Isomorphisms are essential in group theory, graph theory, and topology.
Do isomorphisms apply to non-mathematical contexts?
Yes, isomorphisms apply conceptually to systems, networks, and other structured models.
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Written by
Fiza RafiqueFiza Rafique is a skilled content writer at AskDifference.com, where she meticulously refines and enhances written pieces. Drawing from her vast editorial expertise, Fiza ensures clarity, accuracy, and precision in every article. Passionate about language, she continually seeks to elevate the quality of content for readers worldwide.
Co-written by
Maham Liaqat
