In this paper we introduce the notion of coalgebra symmetry for discrete systems. With concept prove that all radially symmetric systems in standard form are quasi-integrable and variational quasi-radially Poincar\'e--Lyapunov--Nekhoroshev maps order $N-2$, where $N$ degrees freedom system. We also discuss integrability properties several vector which generalisations well-known one degree integ...

We reformulate the bar recursion and induction principles in terms of recursive and wellfounded coalgebras. Bar induction was originally proposed by Brouwer as an axiom to recover certain classically valid theorems in a constructive setting. It is a form of induction on nonwellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function ...

We express the classical p-adic integers Ẑp, as a metric space, as the final coalgebra to a certain endofunctor. We realize the addition and the multiplication on Ẑp as the coalgebra maps from Ẑp × Ẑp.

General logic programs with negation have the 3-valued minimal Herbrand models based on the Kripke’s fixpoint knowledge revision operator and on Clark’s completion. Based on these results we deifine a new algebra , (with the relational algebra embedded in it), and present an algorithmic transformation of logic programs into the system of tuple-variable equations which is a -coalgebra. The solut...

We use prime coalgebras as a generalization of simple coalgebras, and observe that prime subcoalgebras represent the structure of the coalgebra in a more efficient way than simple coalgebras. In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras. It is well known th...

Recently, Turaev [19, 20] (see also Le and Turaev [8] and Virelizier [23]) generalized the notion of a TQFT and Reshetikhin-Turaev invariants to the case of 3-manifolds endowed with a homotopy classes of maps to K(π, 1), where π is a group. One of the key points in [19] is the notion of a crossed Hopf π-coalgebra, here called a Turaev coalgebra or, briefly, a T-coalgebra (see Section 1). As one...

The notion of vertex operator coalgebra is presented which corresponds to the family of correlation functions modeling one string propagating in space-time splitting into n strings in conformal field theory. This notion is in some sense dual to the notion of vertex operator algebra. We prove that any vertex operator algebra equipped with a non-degenerate, Virasoro preserving, bilinear form give...

In a recent paper [1], a systematic construction of integrable Hamiltonians with coalgebra symmetry has been proposed. Such procedure can be applied to any Poisson coalgebra (A,∆) with generators Xi, i = 1, . . . , l and Casimir element C(X1, . . . , Xl) as follows. Let us consider the N -th coproduct ∆(Xi) of the generators and the m-th order (2 ≤ m ≤ N) coproducts ∆(m)(C) of the Casimir opera...

Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.