homomorphism

• “However F also maps functions to homomorphisms, mapping f to its unique extension as a homomorphism, while U maps homomorphisms to functions, namely the homomorphism itself as a function.”

  Algebra

• “We can define an equivalence relation on the domain of a product C, and then take a structure D whose elements are the equivalence classes; the predicate symbols are interpreted in D so as to make the natural map from dom (C) to dom (D) a homomorphism.”

  First-order Model Theory

• “If it is onto, then the inverse map from dom (B) to dom (A) is also a homomorphism, and both the embedding and its inverse are said to be isomorphisms.”

  First-order Model Theory

• “Then T has a model A with the property that for every model B of T there is a unique homomorphism from”

  First-order Model Theory

• “A homomorphism from structure A to structure B is a function f from dom (A) to dom (B) with the property that for every atomic formula”

  First-order Model Theory

• “Next, two properties of that system are established: The rational numbers can be embedded into it, in a way that respects the order and the arithmetic operations defined on those numbers (a corresponding field homomorphism exists); and the new system is continuous, or line-complete, with respect to its order.”

  Dedekind's Contributions to the Foundations of Mathematics

• “Each of these functions is from a generator set to an algebra and therefore has a unique extension to a homomorphism.”

  Algebra

• “Likewise the homomorphism from G to G is an identity function.”

  Algebra

• “Moreover, this correspondence is functorial: any Boolean homomorphism is sent to a continuous map of topological spaces, and, conversely, any continuous map between the spaces is sent to a Boolean homomorphism.”

  Category Theory

• “The key idea is that compositionality requires the existence of a homomorphism between the expressions of a language and the meanings of those expressions.”

  Compositionality

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