Abstract. We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Ge...

We show how coalgebras can be presented by operations and equations. This is a special case of Linton’s approach to algebras over a general base category X , namely where X is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties ...

In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in Pos} posets, with action preserving monotone maps between them, some categorical properties are considered. particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers etc. category. Also, several kinds epimorphisms monomorphisms characterized ...

A concept of equation morphism is introduced for every endofuctor F of a cocomplete category C. Equationally defined classes of F–algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over C, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for “Set–like” categories. By dualizing, we arrive...

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always an algebraic model. Based on this idea, we show local systems connected finite loop space simple When arises from group, recovers special case result Pol and Williamson about cofree $G$-spectra. More generally, if $K$ is closed subgroup $G$ such Weyl $W_GK$ connected, then certain $G$-...

This paper presents a modular framework for the specification of certain inductivelydefined coalgebraic types. Modal logics for coalgebras of polynomial endofunctors on the category of sets have been studied in [16,8]. These logics are here generalised to endofunctors on categories of sorted sets, in order to allow collections of interrelated types to be specified simultaneously. The inductive ...

A propositional logic program P may be identified with a PfPf -coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf )-coalgebra, where C(PfPf ) is the cofree comonad on PfPf , describes derivations by resolution. Using lax semantics, that correspondence may be extended to a class of first-order logic programs without existential variables. The resulting extension...