• In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras...
    20 KB (2,899 words) - 15:17, 30 March 2025
  • also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more...
    23 KB (4,161 words) - 17:18, 1 February 2025
  • algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space...
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  • In mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra...
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  • mathematics, specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with...
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  • measuring coalgebra of two algebras A and B is a coalgebra enrichment of the set of homomorphisms from A to B. In other words, if coalgebras are thought...
    3 KB (407 words) - 01:50, 3 December 2023
  • a coinductive type denotes the assignment of a coalgebra to its unique morphism to the final coalgebra of an endofunctor. These objects are used in functional...
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  • x\\F(f)&=\langle \mathrm {id} _{A},f\rangle \end{aligned}}} The final F-coalgebra ν F {\displaystyle \nu F} has the following morphism associated with it:...
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  • comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. Let K be a field, and C be a coalgebra over K. A (right)...
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  • structure of a coalgebra. There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed...
    31 KB (4,261 words) - 10:53, 26 May 2025
  • Bialgebra (category Coalgebras)
    which is both a unital associative algebra and a counital coassociative coalgebra.: 46  The algebraic and coalgebraic structures are made compatible with...
    12 KB (1,612 words) - 10:08, 11 April 2024
  • list object, respectively. Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for...
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  • Thumbnail for F-algebra
    serve to encapsulate the induction principle, and the dual construction F-coalgebras. If C {\displaystyle C} is a category, and F : C → C {\displaystyle F:C\rightarrow...
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  • simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that...
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  • {\alpha } q\,\}} . In other words, a labelled state transition system is a coalgebra for the functor P ( Λ × − ) {\displaystyle P(\Lambda \times {-})} . There...
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  • (termination), in computer program termination analysis Measuring coalgebra, a coalgebra constructed from two algebras Measure (Apple), an iOS augmented...
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  • integer multiple of another element in the lattice Primitive element (coalgebra), an element X on which the comultiplication Δ has the value Δ(X) = X⊗1...
    832 bytes (140 words) - 08:53, 23 April 2020
  • In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies μ ( x ) = x ⊗ g + g ⊗ x {\displaystyle \mu (x)=x\otimes...
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  • publications indexed by Google Scholar Silva, Alexandra (2010). Kleene coalgebra (PDF). ru.nl (PhD thesis). 694079062. hdl:2066/83205. OCLC 694079062....
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  • unary involution operation Von Neumann algebra (or W*-algebra) See also coalgebra, the dual notion. A different class of "algebras" consists of objects...
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  • Thumbnail for Exterior algebra
    {\displaystyle {\textstyle \bigwedge }(V)} ⁠, giving the structure of a coalgebra. The coproduct is a linear function ⁠ Δ : ⋀ ( V ) → ⋀ ( V ) ⊗ ⋀ ( V )...
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  • sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative ring Rational series Term algebra...
    6 KB (915 words) - 01:13, 27 September 2024
  • From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism followed by an catamorphism: Hylomorphism...
    3 KB (305 words) - 14:41, 6 October 2024
  • usual tensor product are important and widely studied under the name of coalgebras. The notion of monad was invented by Roger Godement in 1958 under the...
    31 KB (4,489 words) - 19:57, 5 July 2025
  • Thumbnail for Monoid (category theory)
    product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra. For any category C, the category [C, C] of its endofunctors has a monoidal...
    5 KB (511 words) - 22:41, 17 March 2025
  • mathematics, the coidentity of a dual group object or the counit of a coalgebra In sociology, double consciousness Double Identity (disambiguation) Secret...
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  • functions. It is both free as an algebra and graded-cofree as a graded coalgebra, so is in some sense as far as possible from being either commutative...
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  • Thumbnail for Lie algebra
    cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie operad Particle physics and representation theory Orthogonal symmetric...
    62 KB (10,497 words) - 10:18, 26 June 2025
  • The reduced tensor coalgebra is the universal cocomplete coalgebra over V {\displaystyle V} , i.e., for any other cocomplete coalgebra C ′ {\displaystyle...
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  • _{x\leq z\leq y}f(x,z)g(z,y).} There is also a definition of incidence coalgebra. In theoretical physics a locally finite poset is also called a causal...
    1 KB (152 words) - 03:18, 13 May 2024