We extend Barr's well-known characterization of the final coalgebra of a Set-endofunctor H as the completion of its initial algebra to the Eilenberg-Moore category Alg(M) of algebras associated to a Set-monad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T, H) with respect to a monad M and… (More)
Motivation Most of coalgebraic logic is focussed on Set-coalgebras and their associated (Boolean) logics. Investigation of coalgebraic logic over Poset already started – expressivity results [Kurz-Kapulkin-Velebil CMCS2010]. Would deserve a systematic investigation of Poset-functors and their coalgebras. In this talk: we restrict on how to move from… (More)
Background Modal logic atomic propositions (a ∧ b) = a ∧ b, = ♦(a ∨ b) = ♦a ∨ ♦b, ♦⊥ = ⊥ (⇒ modal operators are monotone) a ∧ ♦b ≤ ♦(a ∧ b) (a ∨ b) ≤ ♦a ∨ b
If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we prove that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A H. We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.
The notion of crossed product with a coquasi-Hopf algebra H is introduced and studied. The result of such a crossed product is an algebra in the monoidal category of right H-comodules. We give necessary and sufficient conditions for two crossed products to be equivalent. Then, two structure theorems for coquasi Hopf modules are given. First, these are… (More)
The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem () is proven for coquasi-Hopf algebras with bijective antipode. As an application, we generalize… (More)
We show that for a commutative quantale V every functor Set −→ V-cat has an enriched left-Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric. 1 Introduction Coalgebras for a functor T : Set −→ Set capture a wide variety of dynamic… (More)
The distribution of prime numbers is here considered. We show a formula for li −1 and we study the π(x) function and Riemann's hypothesis.
A certain generalisation of the hierarchy of mKdV equations (modified KdV), which forms an integrable system, is studied here. This generalisation is based on a Lax operator associated to the equations, with principal components of degrees between −3 and 0. The results are the following ones: 1) an isomorphism between the space of jets of the system and a… (More)
We study a generalization of the hierarchy of mKdV equations (modi-ed KdV), which forms an integrable system. This generalization is based on a Lax operator associated to the equations, with principal components of degrees between ?3 and 0. The main result of the study is the com-mutation of the classical integrals of motion. For that purpose, one can… (More)