For a given pre-cubical set ( -set) K with two distinguished vertices 0, 1, we prove that the space P (K) 1 0 of d-paths on the geometric realization of K with source 0 and target 1 is homotopy equivalent to its subspace P t (K) 1 0 of tame d-paths. When K is the underlying -set of a Higher Dimensional Automaton A, tame d-paths on K represent step executions of A. Then, we define the cube chain category of K and prove that its nerve is weakly homotopy equivalent to P (K) 1 0 .

We introduce languages of higher-dimensional automata (HDAs) and develop some of their properties. To this end, we define a new category of precubical sets, uniquely naturally isomorphic to the standard one, and introduce a notion of event consistency. HDAs are then finite, labeled, event-consistent precubical sets with distinguished subsets of initial and accepting cells. Their languages are sets of interval orders closed under subsumption; as a major technical step, we expose a bijection between interval orders and a subclass of HDAs. We show that any finite subsumption-closed set of interval orders is the language of an HDA, that languages of HDAs are closed under binary unions and parallel composition, and that bisimilarity implies language equivalence.

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also -sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set K , a CW-complex W (K ) w v that is homotopy equivalent to the space of directed paths between given vertices v, w of K . This construction is functorial with respect to K , and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of W (K ) w v are provided.

Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.

We compute the homology of the spaces of directed paths on a certain class of cubical subcomplexes of the directed Euclidean space R n by a recursive process. We apply this result to calculate the homology and cohomology of the space of directed loops on the (n − 1)-skeleton of the directed torus T n. Keywords Directed paths • Cubical complex • Path space • Homology • Cohomology Mathematics Subject Classification (2000) 55P10 • 55P15 • 55U10 • 68Q85 1 Introduction One of the most important problems of directed algebraic topology is the determination of the homotopy type of spaces of directed paths P(X) y x between two points x, y of a directed space X. This problem seems to be difficult in general; however several Dedicated to Hvedri Inassaridze on his 80th birthday Communicated by Ronald Brown. The authors gratefully acknowledge support from the European Science Foundation network "Applied and Computational Algebraic Topology" that allowed them to collaborate on this paper during mutual visits made possible by grants 4671 and 5432.