Abstract Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in compositional way. In order obtain general framework, need systematic way lift functors: show that the Wasserstein lifting functor, introduced previous work, ...

We study the mathematical structure of notion measurement space, which extends aspects noncommutative topology that are based on quantale theory. This yields a geometric model physical measurements provides realist picture, yet also operational, such and classical information arise interdependently as primitive concepts. A derived observer caters for formulation Bohr's classical/quantum divide....

In this paper, we introduce the notion of {bf K}-flat projective fuzzy quantales, and give an elementary characterization in terms of a fuzzy binary relation on the fuzzy quantale. Moreover, we  prove that {bf K}-flat projective fuzzy quantales are precisely the coalgebras for a certain comonad on the category of fuzzy quantales. Finally, we present two special cases of {bf K} as examples.

This paper attempts to generalize universal algebras on classical sets to $L$-sets when $L$ is a GL-quantale. Some basic notions of fuzzy universal algebra on an $L$-set are introduced, such as subalgebra, quotient algebra, homomorphism, congruence, and direct product etc. The properties of them are studied. $L$-valued power algebra is also introduced and it is shown there is an onto homomorphi...

Behavioural preorders and equivalences for higher-order languages have been extensively studied in the last decades, often leading to rich and satisfactory theories. However, in presence of effectful computations ordinary behavioural relations can be too discriminating, as highlighted by probabilistic higher-order languages. An elegant way to avoid such problem is to move from qualitative, bool...

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations ...

The development of operational quantum logic points out that classical boolean structures are too rigid to describe the actual and potential properties of quantum systems. Operational quantum logic bears upon basic axioms which are motivated by empirical facts and as such supports the dynamic shift from classical to non-classical logic resulting into a dynamics of logic. On the other hand, a dy...

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloidenriched category theory. We discuss at length the definitions of quantaloid, quantaloidenriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indi...

The starting point of the present study is the interpretation of intuitionistic linear logic in Petri nets proposed by U. Engberg and G. Winskel. We show that several categories of order algebras provide equivalent interpretations of this logic, and identify the category of the so called strongly coherent quantales arising in these interpretations. The equivalence of the interpretations is inti...

We use continuity spaces, a common refinement of posets and metric spaces, to develop a general theory of semantic domains which includes metric spaces and domains of cpo’s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming. Our proposal for a general notion ...