Nonassociative coalgebras

Dual Coalgebras of Jordan Bialgebras and Superalgebras

“…As shown in [9], a subcoalgebra of A, generated by a subset S, coincides with the A * -subbimodule generated by S. Therefore, a coalgebra (A, Δ) is locally finite-dimensional if and only if A is a locally finite-dimensional A * -bimodule. Denote by Loc(A) the sum of all finite-dimensional subcoalgebras of A.…”

“…Now, let A • be the sum of all good subspaces of A * . Then A • is the greatest good subspace, and so A • is a coalgebra with comultiplication Δ • = Δ A • (see [3,9]). The coalgebra (…”

“…Denote by Loc(A) the sum of all finite-dimensional subcoalgebras of A. We have Theorem (ACM) 1 [9]. Let A be an algebra over a field F .…”

“…In [9], the following definition of coalgebra is given that relates to some variety of algebras. Definition.…”

Recognizable Formal Series on Trees and Cofree Coalgebraic Systems

“…The appropriate notion of coalgebra here is that of F--coalgebra (or, for short, -coalgebra, which will always mean F--coalgebra in this paper), which we define to be a vector space C with a linear map ω C → ⊗ p C for each ω in p . This definition of -coalgebras includes the case of noncoassociative coalgebras, that is, the binary case, with = 2 = (see, for example, [Gr1,ACM,AC,M]), and of course includes the classical coassociative counital case in which = 2 ∪ 0 = ∪ ε . An important distinction from the classical case (already known in the binary case [M], [Gr2]) is that -coalgebras need not be locally finite (that is, a finitely generated subcoalgebra need not be finite-dimensional).…”