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Maths › 03. Logic and Foundations ›

4.4 Isomorphism

Structures are often presented using particular sets, symbols, and operations, but many mathematical questions concern the structure itself rather than the specific names of its elements. Isomorphism formalizes this idea by identifying structures that have the same internal organization, even when their underlying domains are different. Definition 4.31 (Isomorphism) Let $\mathcal A$ and $\mathcal B$ be structures for the same signature. An isomorphism from $\mathcal A$ to $\mathcal B$ is...
logic model theory isomorphism invariants May 2, 2026
Maths › 00. General Mathematics ›

04.4 Invariants

4.4 Invariants and Classification An invariant is a property, quantity, or structure that remains unchanged under a chosen class of transformations. Invariants are central to structural thinking. They allow us to compare objects without depending on representation. If two objects are isomorphic, their structural invariants agree. Therefore, if an invariant differs, the objects cannot be isomorphic. For example, two finite sets with different cardinalities cannot be bijective. Two vector spaces...
mathematics structure invariants classification isomorphism May 2, 2026
Maths › 00. General Mathematics ›

04.3 Isomorphism

4.3 Isomorphism as Sameness Isomorphism is the mathematical notion of structural sameness. Two objects are isomorphic when there is a reversible map between them that preserves the relevant structure. Equality is usually too strict. Two objects can have different representations and still behave identically for the purposes of a theory. For example, the additive group of integers and the additive group of even integers are not equal as sets: $$...
mathematics structure isomorphism equivalence classification May 2, 2026