Universal homogeneous causal sets.
Journal of Mathematical Physics 46 (2005), 122503 1-10
Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete space-time in quantum gravity. We show that the class C of all countable past-finite causal sets contains a unique causal set (U,<) which is universal (i.e., any member of C can be embedded into (U,<) and homogeneous (i.e., (U,<) has maximal degree of symmetry). Moreover, (U,<) can be constructed both probabilistically and explicitly.
Ordinal scales in the theory of measurement.
J. Math. Psychol. 31 (1987), 60 - 82.
We introduce a new comparison criterion for scales of weakly ordered sets, and we obtain a complete characterization of the structure of the system S(A) of all equivalence classes of scales of a weakly ordered set (A,<) under this relation. In particular, it is shown that this system S(A) is a lattice and has indeed a very rich structure theory. We also consider a measure-theoretic concept for extensions of scales of some structure to scales of a larger structure, and we apply our results to two classes of relational structures studied in the literature. We also characterize when an order-preserving function $f: A\mapsto R$, defined on an arbitrary subset A of R, can be extended to an order-preserving function defined on all of R, or even to an order-automorphism of (R,<); this characterization provides the basis for our results on scales of weakly ordered sets.
Classification and transformation of ordinal scales in the theory of measurement.
in: "Progress in Mathematical Psychology I" (E. Roskam, R. Suck, eds.)
North Holland, Amsterdam. 1987, 47 - 55.
We introduce a new comparison criterion for scales of weakly ordered sets, and we obtain a complete characterization of the structure of the system S(A) of all equivalence classes of scales of a weakly ordered set (A,<) under this relation. In particular, it is shown that this system S(A), which determines the transformation behaviour of the scales of (A,<), is a lattice and in most cases even a power set Boolean algebra.
Uniqueness of semicontinuous ordinal utility functions.
J. of Economics, suppl. 8 (1999), 23 - 38.
We introduce a comparison criterion for semicontinuous utility functions of weakly ordered sets, and we show that the collection S(A) of all equivalence classes of semicontinuous utility functions of a weak order (A,<) becomes a partially ordered set under this relation. Its structure can be characterized by order-theoretic properties of the given weak order (A,<). We also consider a concept for consistent extensions of utility functions from some structure A to a larger one.