joint embedding property

Let K be a class of models (structuresMathworldPlanetmath) of a given signaturePlanetmathPlanetmathPlanetmath. We say that K has the joint embedding property (abbreviated JEP) iff for any models A and B in K there exists a model C in K such that both A and B are embeddable in C. [1, 2]

0.0.1 Examples

Examples include [2]:

  • The class of all groups.

  • The class of all monoids.

  • The class of all non-trivial Boolean algebrasMathworldPlanetmath.

As is the case with the above examples, classes having the joint embedding property often satisfy an even stronger condition - for every indexed family of models in the class there is a model in the class into which each member of the family can be embedded. This is known as the strong joint embedding property (abbreviated SJEP). [3]

In general any factor embeddable class closed underPlanetmathPlanetmath productsMathworldPlanetmathPlanetmath will have the strong joint embedding property. [2]

0.0.2 Characterizations

Elementary classes with the joint embedding property may be characterized syntactically and semantically:

Let T be a first order theory in a languagePlanetmathPlanetmath L and let K be the class of models of T then:

  1. 1.

    K has the joint embedding property iff for all universal sentences ϕ and ψ in L, Tϕψ implies either Tϕ or Tψ. [1]

  2. 2.

    If T is consistent, then K has the joint embedding property iff T has an ultra-universal model. [2]


  • 1 Abraham Robinson: ForcingMathworldPlanetmath in model theoryMathworldPlanetmath, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 245-250
  • 2 Colin Naturman, Henry Rose: Ultra-universal models, Quaestiones Mathematicae, 15(2), 1992, 189-195
  • 3 Colin Naturman: Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Mathematics, 1991
Title joint embedding property
Canonical name JointEmbeddingProperty
Date of creation 2013-03-22 19:36:14
Last modified on 2013-03-22 19:36:14
Owner Naturman (26369)
Last modified by Naturman (26369)
Numerical id 25
Author Naturman (26369)
Entry type Definition
Classification msc 03C52
Synonym JEP
Synonym SJEP
Related topic ultrauniversal
Related topic amalgamationproperty
Related topic UltraUniversal
Related topic FactorEmbeddable
Defines joint embedding property
Defines strong joint embedding property