In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras...

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...

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...

mathematics, specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with...

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...

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...

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...

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:...

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...

which is both a unital associative algebra and a counital coassociative coalgebra.: 46  The algebraic and coalgebraic structures are made compatible with...

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)...

list object, respectively. Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for...

(termination), in computer program termination analysis Measuring coalgebra, a coalgebra constructed from two algebras Measure (Apple), an iOS augmented...

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...

{\alpha } q\,\}} . In other words, a labelled state transition system is a coalgebra for the functor P ( Λ × − ) {\displaystyle P(\Lambda \times {-})} . There...

simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that...

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...

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...

publications indexed by Google Scholar Silva, Alexandra (2010). Kleene coalgebra (PDF). ru.nl (PhD thesis). 694079062. hdl:2066/83205. OCLC 694079062....

unary involution operation Von Neumann algebra (or W*-algebra) See also coalgebra, the dual notion. A different class of "algebras" consists of objects...

sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative ring Rational series Term algebra...

{\displaystyle {\textstyle \bigwedge }(V)} ⁠, giving the structure of a coalgebra. The coproduct is a linear function ⁠ Δ : ⋀ ( V ) → ⋀ ( V ) ⊗ ⋀ ( V )...

From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism followed by an catamorphism: Hylomorphism...

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...

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...

mathematics, the coidentity of a dual group object or the counit of a coalgebra In sociology, double consciousness Double Identity (disambiguation) Secret...

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...

The reduced tensor coalgebra is the universal cocomplete coalgebra over V {\displaystyle V} , i.e., for any other cocomplete coalgebra C ′ {\displaystyle...

cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie operad Particle physics and representation theory Orthogonal symmetric...

all maps from the morphisms of C to R, denoted F(C), and has a natural coalgebra structure. Thus for a locally finite category, the dual of a category...