We combine two research directions of the past decade, namely the development of a lax-algebraic framework for categories of interest to topologists and analysts, and the exploration of key topological concepts, like separation and compactness, in an abstract category which comes equipped with an axiomatic notion “closed” or “proper” map. Hence, we present various candidates for such notions in...

Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...

There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra g̃l(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves. Under some generic restrictions on the structure of the Lax matrix, we propose an algebrai...

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...

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...

In this paper we prove that the generalization to RP of the pentagram map defined in [4] is invariant under certain scalings for any n. This property allows the definition of a Lax representation for the map, to be used to establish its integrability.

Two models of classical linear logic are set up. First our recent version of AJM games model which will be our source model. Then the target model, polarized pointed relations, a variant of the plain rela-tional model which is constructed in two steps: rst the model of pointed relations, then the additional polarization structure which yields a proper duality. Then the natural time-forgetting m...

Let B be the unit ball in the complex space C. Write Rat(B,B) for the space of proper rational holomorphic maps from B into B and Poly(B,B ) for the set of proper polynomial holomorphic maps from B into B . We say that F and G ∈ Rat(B,B) are equivalent if there are automorphisms σ ∈ Aut(B) and τ ∈ Aut(B ) such that F = τ ◦G◦σ. Proper rational holomorphic maps from B into B with N ≤ 2n− 2 are eq...