Let T be a first order theory. A model M of T is said to be an ultra-universal model of T iff for every model A of T there exists and ultra-power of M into which A can be embedded. [1, 3]

If T has an ultra-universal model it is referred to as an ultra-universal theory. The class of models of an ultra-universal theory is called an ultra-universal class. If T is an ultra-universal theory with elementary class K and ultra-universal model M then M is said to be ultra-universal in K. [3]

0.0.1 Characterizations

Ultra-universal classes are precisely the non-empty elementary classes having the joint embedding property. [3]

Ultra-universal models can be characterized in terms of universalPlanetmathPlanetmathPlanetmath or existential sentences:

Let T be theory and let M be a model of T. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath: [3]

  1. 1.

    M is an ultra-universal model of T

  2. 2.

    Every universal sentence holding in M holds in all models of T

  3. 3.

    Every existential sentence holding in some model of T holds in M

A theory T is ultra-universal iff it is consistent and for all universal sentences ϕ and ψ, Tϕψ implies Tϕ or Tψ. [3]

A completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath consistent theory is always ultra-universal. More generally the set of universal sentences Σ of a complete consistent theory T is always an ultra-universal theory - a model of T is an ultra-universal model of Σ. Ultra-universal theories are precisely those theories T which are consistent and can be extended to a complete consistent theory without introducing any universal sentences that are not deducibleMathworldPlanetmath from T. [3]

In terms of the Lindenbaum-Tarski algebra for a first order language L, a theory T in L is ultra-universal iff the filter F that it generates in the Lindenbaum-Tarski algebra is proper and can be extended to an ultrafilterMathworldPlanetmathPlanetmath U such that FA=UA where A is the sub-lattice of universal sentences. Moreover T is ultra-universal iff FA is a prime proper filter in A. Thus ultra-universal theories correspond to prime proper filters in the boundedPlanetmathPlanetmathPlanetmathPlanetmath distributive latticeMathworldPlanetmath of universal sentences. [3]

0.0.2 Examples


  • 1 Peter Bruyns, Henry Rose: Varieties with cofinalPlanetmathPlanetmath sets: examples and amalgamation, Proc. Amer. Math. Soc. 111 (1991), 833-840
  • 2 Colin Naturman, Henry Rose: Interior algebras: some universal algebraic aspects, J. Korean Math. Soc. 30 (1993), No. 1, pp. 1-23
  • 3 Colin Naturman, Henry Rose: Ultra-universal models, Quaestiones Mathematicae, 15(2), 1992, 189-195
Title ultra-universal
Canonical name Ultrauniversal
Date of creation 2013-03-22 19:36:18
Last modified on 2013-03-22 19:36:18
Owner Naturman (26369)
Last modified by Naturman (26369)
Numerical id 29
Author Naturman (26369)
Entry type Definition
Classification msc 03C20
Classification msc 03C52
Classification msc 03C50
Related topic universal
Related topic jointembeddingproperty
Related topic JointEmbeddingProperty
Defines ultra-universal model
Defines ultra-universal theory
Defines ultra-universal class