A bisimulation for a coalgebra of functor on the category sets can be described via in relations, lifted functor. final then gives rise to coinduction principle, which states that two bisimilar elements are equal. For polynomial functors, this leads well-known descriptions. In present paper we look at dual notion "apartness". Intuitively, apart if there is positive way distinguish them. Phrased...

Motivated by Dennis Dams’s studies of overand underapproximation of statetransition systems, we define a logical-relation calculus for Galois-connection building. The calculus lets us define overapproximating Galois connections in terms of lower powersets and underapproximating Galois connections in terms of upper powersets. Using the calculus, we synthesize Dams’s most-precise overand underapp...

Theorem (5) states that if an iterate of a function has a unique xpoint then it is also the xpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory. Next, we prove th...

This paper explores the relationship between simulation and the emerging topics of Lean Thinking and lean construction. The ideas which have motivated the evolution of lean construction concepts are presented. The use of simulation as a means of evaluating the benefit of using lean construction techniques is discussed. Two examples of lean thinking based re-design of common construction process...

Liftings of endofunctors on sets to relations are commonly used capture bisimulation coalgebras. Lax versions have been in those cases where strict lifting fails bisimilarity, as well modeling other notions simulation. This paper provides tools for defining and manipulating lax liftings. As a central result, we define notion distributive law functor over the powerset monad, show that there is a...

the purpose of this paper is to introduce the concept of l-fuzzybilinear operators. we obtain a decomposition theorem for l-fuzzy bilinearoperators and then prove that a l-fuzzy bilinear operator is the same as apowerset operator for the variable-basis introduced by s.e.rodabaugh (1991).finally we discuss the continuity of l-fuzzy bilinear operators.

This paper demonstrates the redundancies concerning the increasing popular ``soft set" approaches to general topologies. It is shown that there is a complement preserving isomorphism (preserving arbitrary $widetilde{bigcup}$ and arbitrary $widetilde{bigcap}$) between the lattice ($mathcal{ST}_E(X,E),widetilde{subset}$) of all soft sets on $X$ with the whole parameter set $E$ as domains and the ...

Set functions are (or signals) indexed by the powerset (set of all subsets) a finite set N. They fundamental and ubiquitous in many application domains have been used, for example, to formally describe or quantify loss semantic image segmentation, informativeness sensors sensor networks utility sets items recommender systems, cooperative games game theory, bidders combinatorial auctions. In par...

The paper investigates non-deterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing mac...