同型性 | 群論、代数構造、同値関係

Isomorphism | Group Theory, Algebraic Structures, Equivalence Relations | Britannica Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos isomorphism Introduction References & Edit History Related Topics Images Contents CITE verified Cite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style MLA APA Chicago Manual of Style Copy Citation Share Share Share to social media Facebook X URL https://www.britannica.com/science/isomorphism-mathematics Feedback External Websites Feedback Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). Feedback Type Select a type (Required) Factual Correction Spelling/Grammar Correction Link Correction Additional Information Other Your Feedback Submit Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. External Websites Emory University - Department of Mathematics - Isomorphisms and Composition Homomorphisms and isomorphisms CORE - Testing isomorphism of graded algebras Mathematics LibreTexts - Isomorphisms University of Maryland - Department of Mathematics - Isomorphisms Simon Fraser Uniersity - Isomorphism Frontiers - Frontiers in Psychology - The Importance of Isomorphism for Conclusions about Homology: A Bayesian Multilevel Structural Equation Modeling Approach with Ordinal Indicators CiteSeerX - A First-Order Isomorphism Theorem (PDF) Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. Isomorphism - Student Encyclopedia (Ages 11 and up) isomorphism mathematics Ask Anything Homework Help Written by William L. Hosch William L. Hosch was an editor at Encyclopædia Britannica. William L. Hosch Fact-checked by Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... Britannica Editors History Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything isomorphism , in modern algebra , a one-to-one correspondence ( mapping ) between two sets that preserves binary relationships between elements of the sets. For example, the set of natural numbers can be mapped onto the set of even natural numbers by multiplying each natural number by 2. The binary operation of adding two numbers is preserved—that is, adding two natural numbers and then multiplying the sum by 2 gives the same result as multiplying each natural number by 2 and then adding the products together—so the sets are isomorphic for addition. In symbols, let A and B be sets with elements a n and b m , respectively. Furthermore, let ⊕ and ⊗ indicate their respective binary operations, which operate on any two elements from a set and may be different. If there exists a mapping f such that f ( a j ⊕ a k ) = f ( a j ) ⊗ f ( a k ) and its inverse mapping f −1 such that f −1 ( b r ⊗ b s ) = f −1 ( b r ) ⊕ f −1 ( b s ), then the sets are isomorphic and f and its inverse are isomorphisms. If the sets A and B are the same, f is called an automorphism . Related Topics: homomorphism (Show more) See all related content Because an isomorphism preserves some structural aspect of a set or mathematical group , it is often used to map a complicated set onto a simpler or better-known set in order to establish the original set’s properties. Isomorphisms are one of the subjects studied in group theory . William L. Hosch