Definition & Meaning | English word ADJUNCTION

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Definitions of ADJUNCTION

1. The act of joining; the thing joined or added.
2. (legal) The joining of personal property owned by one to that owned by another.
3. (mathematics, chiefly, algebra and number theory) The process of adjoining elements to an algebraic structure (usually a ring or field); the result of such a process.
4. (category theory, loosely) A relationship between a pair of categories that makes the pair, in a weak sense, equivalent.
5. (category theory, strictly) A natural isomorphism between a pair of functors satisfying certain conditions, whose existence implies a close relationship between the functors and between their (co)domains; the natural isomorphism, functors, and their (co)domains thought of as a single object.

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Examples of Using ADJUNCTION in a Sentence

• Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic.
• In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories.
• His contributions in the subsequent 35 years of his career include transfer grammar, string analysis (adjunction grammar), elementary sentence-differences (and decomposition lattices), algebraic structures in language, operator grammar, sublanguage grammar, a theory of linguistic information, and a principled account of the nature and origin of language.
• The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
• In the category of abelian groups, pushouts can be thought of as "direct sum with gluing" in the same way we think of adjunction spaces as "disjoint union with gluing".
• An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D.
• Monadic, an adjunction if and only if it is equivalent to the adjunction given by the Eilenberg–Moore algebras of its associated monad, in category theory.
• The adjunction is thus akin to currying, taking maps on cartesian products to their curried form, and is an example of Eckmann–Hilton duality.
• The existence theorem for the twisted inverse image is the name given to the proof of the existence for what would be the counit for the comonad of the sought-for adjunction, namely a natural transformation.
• Q is interpretable in a fragment of Zermelo's axiomatic set theory, consisting of extensionality, existence of the empty set, and the axiom of adjunction.
• Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.
• Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations.
• Instead, disparities in grammaticality emerge in environments of topicalization and adverbial adjunction.
• 1933 – Belarusian Polytechnic Institute after adjunction of Horki Institute of Land Amelioration and Minsk Institutions of Peat, Chemical Engineering, Civil Engineering, Electrical Engineering and Food Industry.

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