Applications of ordered Structures - Applications of ordered Structures - Applications of ordered Structures - Applications of ordered Structures - Applications

**
Universal homogeneous causal sets.
**

*Journal of Mathematical Physics* 46 (2005), 122503 1-10

__Abstract.__

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.

__Abstract.__

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.

__Abstract.__

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.

__Abstract.__

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.