Vol 55, No 3 (2016)

This is the third in a three-part series of papers shortly denoted by Part I [1], and Part II [2], and Part III. The papers mentioned are devoted to some Whiteheadean theories of space and time. Part I contains a historical introduction and some facts from static mereotopology. Part II introduces a point-based definition of a dynamic model of space and a definition of a standard dynamic contact algebra based on the so-called snapshot construction. The given model has an explicit time structure with an explicit set of time points equipped with a before–after relation and a set of regions changing in time, called dynamic regions. The dynamic model of space contains several definable spatiotemporal relations between dynamic regions: space contact, time contact, precedence, and some others. In Part II, a number of statements for these relations are proven, which in the present Part III are taken as axioms for the abstract definition of some natural classes of dynamic contact algebras, considered as an algebraic formalization of dynamic mereotopology. Part III deals with a representation theory for dynamic contact algebras, and the main theorem says that each dynamic contact algebra in some natural class is representable as a standard dynamic contact algebra in the same class.

We give sufficient conditions under which an upper semilattice of computably enumerable m-degrees is isomorphic to an ideal of a Rogers semilattice of a two-element family of sets in the Ershov hierarchy. It is shown that the given conditions are not necessary.

We introduce a class of existentially Steinitz structures containing, in particular, the fields of real and complex numbers. A general result is proved which implies that if \( \mathfrak{M} \) is an existentially Steinitz structure then the following structures cannot be embedded in any structure Σ-presentable with trivial equivalence over ℍ\( \mathbb{F} \)(\( \mathfrak{M} \)): the Boolean algebra of all subsets of ω, its factor modulo the ideal consisting of finite sets, the group of all permutations on ω, its factor modulo the subgroup of all finitary permutations, the semigroup of all mappings from ω to ω, the lattice of all open sets of real numbers, the lattice of all closed sets of real numbers, the group of all permutations of ℝ Σ-definable with parameters over ℍ\( \mathbb{F} \)(ℝ), and the semigroup of such mappings from ℝ to ℝ.