The paper is devoted to the investigation of behavioural equivalences for Petri nets with silent transitions (τ -equivalences). Basic τ -equivalences and back-forth τ -bisimulation equivalences are supplemented by new ones, giving rise to the complete set of equivalence notions in interleaving / true concurrency and linear / branching time semantics. Their interrelations are examined for the ge...

The hierarchy of untimed equivalences is well understood for action-based systems. This is not the case for timed systems, where it is, for example, possible to detect concurrency by single timed action execution. To clarify the connection between equivalences in timed systems, a timed version of configuration structures is introduced together with timed equivalence notions adapted from untimed...

The paper is devoted to the investigation of behavioural equivalences of concurrent systems modelled by Petri nets. Back-forth bisimulation equivalences known from the literature are supplemented by new ones, and their relationship with basic behavioural equivalences is examined for the whole class of Petri nets as well as for their subclass of sequential nets. In addition, the preservation of ...

We propose behavioral specification theories for most equivalences in the linear-time–branching-time spectrum. Almost all previous work on specification theories focuses on bisimilarity, but there is a clear interest in specification theories for other preorders and equivalences. We show that specification theories for preorders cannot exist and develop a general scheme which allows us to defin...

In this article, we study model structures on the category of finite graphs with $\times$-homotopy equivalences as weak equivalences. We show that there does not exist an analogue Str\o{}m-Hurewicz structure graphs. More interestingly, have a whenever class cofibrations is subclass graph inclusions.

In this paper, we first introduce the concepts of G-systems, quotient G-systems and isomorphism theorems on G-systems of n-ary semihypergroups . Also we consider the Green's equivalences on G-systems and further in-vestigate some of their properties. A number of n-ary semihypergroups are constructed and presented as examples in this paper.

In this paper, we consider an enriched orthogonality for classes of spaces, with respect to groupoids, simplicial sets and spaces themselves. This point of view allows one to characterize homotopy equivalences, shape and strong shape equivalences. We show that there exists a class of spaces, properly containing ANR-spaces, for which shape and strong shape equivalences coincide. For such a class...

We introduce a new class of operations called asymmetric equivalences. Several properties of asymmetric equivalence operations have been investigated. Based on the asymmetric equivalence, quasi-metric spaces are constructed on [0, 1]. Finally, we discuss symmetrization of asymmetric equivalences.

We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object in this category, denoted hEndad(X). Since the homotopy theory of ring objects in Sp(sAb) is equivalent to the homotopy theory of differential graded algebras, on...

In a previous paper we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of “relative categories.” Here, we characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the seco...