Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the res...

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....

We revisit sheaves on locales by placing them in the context of the theory of Hilbert quantale modules. The local homeomorphisms p : X → B are identified with the locales X that are Hilbert B-modules equipped with a natural notion of basis. These modules form a full subcategory B-HMB of the category of Hilbert B-modules where all the homomorphisms are adjointable: for each homomorphism h there ...

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...

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically signi…cant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated ...

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 ...

In this paper,  it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying conver...

In this paper, we describe the Grothendieck group $$K_0(V)$$ of an absolute matrix order unit space V. For purpose, discuss direct limit spaces. We show that $$K_0$$ is a functor from category spaces with morphisms as unital completely $$\vert \cdot \vert$$ -preserving maps to abelian groups. study structure in and prove under certain condition, ordered group. also additive on orthogonal maps.