This paper relates labelled transition systems and coalgebras with the motivation of comparing and combining their complementary contributions to the theory of concurrent systems. The well-known mismatch between these two notions concerning the morphisms is resolved by extending the coalgebraic framework by lax cohomomorphisms. Enriching both labelled transition systems and coalgebras with alge...

A new Lax representation for the Bogoyavlensky lattice is found, its r–matrix interpretation is elaborated. The r–matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of asso-ciative algebras. The theory of such nonlocal structures is developed, the Poisson property of the monodromy map is worked out in the most general situation. Some proble...

The characterization of stably closed maps of topological spaces as the closed maps with compact fibres and the role of the Kuratowski-Mrówka’ Theorem in this characterization are being explored in the general context of lax (T, V )-algebras, for a quantale V and a Setmonad T with a lax extension to V -relations. The general results are being applied in standard (topological and metric) and non...

We study the conformal properties of the multi-constraint KP hierarchy and its nonstandard partner by covariantizing their corresponding Lax operators. The associated second Hamiltonian structures turn out to be nonlocal extension of Wn algebra by some integer or half-integer spin fields depending on the order of the Lax operators. In particular, we show that the complicated second Hamiltonian ...

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms idempotents. establish their main properties and interpretation the category theory. The notion minors is generalised matrix. some examples matrices, relations with Lax operators obtain formulae minors. In particular, we consider types $B$, $C$ $D$ introduced by A. Molev relation Brauer algebras. Infinite-d...

Categories of lax (T, V )-algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V )-algebras, with T preserving inverse images. Moreov...

The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. key elements a class loop algebras consisting block matrices blocks the same size. As illustrative examples, isospectral non-isospectral corresponding commutator relations their Lax operators computed explicitly in c...

Beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing nonlinear discrete integrable couplings. Discrete variational identities over the associated loop algebras are used to build Hamiltonian structures for the resulting integrable couplings. We illustrate the application of the scheme by means of an enlargedVolterra spectral problemandp...