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...

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...

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...

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...

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:...

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)...

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

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...

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...

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...

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

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

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

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

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...

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...

_{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...

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...

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...

ISBN 0-521-77920-0. Klaus Denecke; Shelly L. Wismath (2009). Universal Algebra and Coalgebra. World Scientific. pp. 21–23. ISBN 978-981-283-745-5. T.H. Tse (2010)...