Cubical local partial orders on cubically subdivided spaces: existence and construction

Authors:
Lisbeth Fajstrup
Affiliations:
Department of Mathematical Sciences, University of Aalborg, Aalborg Øst, Denmark
Venue:
Theoretical Computer Science - Spatial representation: Discrete vs. continous computational models
Year:
2006

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Abstract

The geometric models of higher dimensional automata (HDA) and Dijkstra's PV-model are cubically subdivided topological spaces with a local partial order. If a cubicalization of a topological space is free of immersed cubic Möbius bands, then there are consistent choices of direction in all cubes, such that any n-cube in the cubic subdivision is dihomeomorphic to [0, 1]n with the induced partial order from Rn. After subdivision once, any cubicalized space has a cubical local partial order. In particular, all triangularized spaces have a cubical local partial order. This implies in particular that the underlying geometry of an HDA may be quite complicated.