in this work, we describe an adjunction between the comma category of set-based monads under the v -powerset monad and the category of associative lax extensions of set-based monads to the category of v -relations. in the process, we give a general construction of the kleisli extension of a monad to the category of v-relations.

We discuss a differential integrable hierarchy, which we call the (N,M)–th KdV hierarchy, whose Lax operator is obtained by properly adding M pseudo–differential terms to the Lax operator of the N–th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi– field representation of KP hierarchy as sub–systems and naturally appears in multi–matrix models. The N + 2M − 1 ...

We develop the approach to the problem of integrable discretization based on the notion of r–matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non–locality. We demonstrate how to overcome this drawback. Namely...

The Toda flows are examples of integrable Hamiltonian systems which are of great interest from the point of view of pure mathematics as well as in applications. They originated as a description of a 1-dimensional lattice of particles with exponential interaction [29]. Flaschka [14] found a Lax pair for the equations and Moser [25] analyzed the dynamics and scattering behavior of the system. Adl...

We investigate a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal veri cation of computer hardware. The logic has emerged from an attempt to express correctness `up to' behavioural constraints | a central notion in hardware veri cation | as a logical modality. As a modal logic it is special since it features a single modal...

A hierarchy of first-degree time-dependent symmetries is proposed for Dirac soliton hierarchy and their commutator relations with time-dependent symmetries are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is elucidated and a Lax operator algebra associated with Virasoro symmetry algebra is given. The main purpose of the present letter is to construct a hierarchy of fir...

We show that morphisms from n homotopy unital A∞-algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A ∞, whose algebras are homotopy unital A∞-algebras. The operad A∞ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A ∞-morphisms with several entries is presented as a convolution of a...