The paper provides a quantitative algebraic analysis of a BB’84-type quantum key distribution protocol. The analysis is done in an algebraic setting, where classical and quantum variables form a module for the quantale formed from the communication and quantum actions. The module-quantale pair is endowed with sup-maps that encode uncertainties of agents involved in the protocol, about the varia...

In previous papers, several $T_{0}$ and Hausdorff objects in topological categories are introduced compared. The main objectives of this paper to characterize $\overline{T_{0}}$, $T_{0}$, $T_{1}$ pre-$\overline{T_{2}}$ the category quantale-valued closure space as well examine their mutual relationship.

We propose various methods for combining or amalgamating propositional languages and deductive systems. make heavy use of quantales quantale modules in the wake previous works by present other authors. also describe quite extensively relationships among algebraic order-theoretic constructions corresponding ones based on a purely logical approach.

When replacing the non-negative real numbers with their addition by a commutative quantale V, under a metric lens one may then view small V-categories as sets that come with a V-valued distance function. The ensuing category V-Cat is well known to be a concrete topological category that is symmetric monoidal closed. In this paper we show which concrete symmetric monoidalclosed topological categ...

The characterization of stably closed maps of topological spaces as the closed maps with compact fibres and the role of the Kuratowski-Mrówka’ Theorem in this characterization are being explored in the general context of lax (T, V )-algebras, for a quantale V and a Setmonad T with a lax extension to V -relations. The general results are being applied in standard (topological and metric) and non...

We consider compatible group structures on a V -category, where is quantale, and we study the topological algebraic properties of such groups. Examples are preordered groups, metric ultrametric probabilistic (ultra)metric In particular, show that, when frame, symmetric -groups satisfy very strong categorical-algebraic properties, typical category form protomodular category.

In this paper, we consider a non-posetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as balanced mon...

We formulate an elementary condition on an involutive quantaloidQ under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawver...

Abstract Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest generic approaches to behavioural distances. In particular, coalgebraic methods capture variations system type (nondeterministic, probabilistic, game-based etc.), and notion of quantale abstracts over actual values take, thus covering,...