In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general cor...

The classical theory of deterministic automata is presented in terms of the notions of homomor-phism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the p...

The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra [3] can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras A = {A α } α∈π , ...

The semantics of process calculi has traditionally been specified by labelled transition systems (LTSs), but with the development of name calculi it turned out that definitions employing reduction semantics are sometimes more natural. Reactive Systems à la Leifer and Milner allow to derive from a reduction semantics definition an LTS equipped with a bisimilarity relation which is a congruence. ...

We give a general framework connecing a branching time relation on nodes of a transition system to a final coalgebra for a suitable endofunctor. Examples of relations treated by our theory include bisimilarity, similarity, upper and lower similarity for transition systems with divergence, similarity for discrete probabilistic systems, and nested similarity. Our results describe firstly how to c...

Gelfand, Retakh, Serconek and Wilson, in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra ...

Traditionally, traces are the sequences of labels associated with paths in transition systems X → P(A×X). Here we describe traces more generally, for coalgebras of the form X → P(F (X)), where F is a polynomial functor. The main result states that F ’s final coalgebra Z ∼= −→ F (Z) gives rise to a weakly final coalgebra with state space P(Z), in a suitable category of coalgebras. Weak finality ...

A quantum sl(2,R) coalgebra (with deformation parameter z) is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the non-constant curvature analogues of the spherical, hyperbolic and (anti-)de Sitter spaces. The connection and curvature tensors for these “deformed” spaces are fully studied by working on two different phase spaces. ...