factor embeddable

Let K be a class of models (structuresMathworldPlanetmath) of a given signaturePlanetmathPlanetmathPlanetmath. Consider a (non-empty) family of structures {Ai:iI} in K. If jI and f:AjiIAi is an embeddingPlanetmathPlanetmathPlanetmath, we say that f is a factor embedding. [1, 2] If additionally f satisfies the condition that πjf is the identityPlanetmathPlanetmathPlanetmathPlanetmath on Ai, where πj:iIAiAj is the jth projection, then f is said to be a strong factor embedding. [2] K is said to be a factor embeddable class iff for every (non-empty) family of structures {Ai:iI} in K and every jI there is a factor embedding f:AjiIAi. [1, 2]

The definition above does not require the productMathworldPlanetmathPlanetmath iIAi to be a member of K, however many interesting examples of factor embeddable classes are in fact closed underPlanetmathPlanetmath products. Factor embeddable classes that are closed under finite products (or equivalently under binary products) have the joint embedding property. Factor embeddable classes closed under arbitrary products have the strong joint embedding property.

0.0.1 Characterization

Factor embeddable classes have an easy to prove but somewhat unintuitive characterization which does not mention the conceptsMathworldPlanetmath of product or embedding:

The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath for a class K of models [2]:

  1. 1.

    K is factor embeddable.

  2. 2.

    For every pair of models A,BK there exists a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f:AB.

To see the above, suppose K is factor embeddable and consider models A,BK. Then there exists a factor embedding from A into the product A×B. Composing this embedding with the projection onto B gives a homomorphism f:AB. Conversely suppose such a homomorphism f:AB exists for all A,BK and consider a family {Ai:iI} in K and jI. We can define a strong factor embedding f:AjiIAi by choosing homomorphisms fi:AjAi for all iI with fj the identity map on Aj, and then for all aA setting f(a)i=fi(a) for each iI. [2]

The above proof shows that the factor embeddings guaranteed to exist for a factor embeddable class can always be chosen to be strong factor emebeddings. [2]

A corollory of the above is that if there exists a model which is a retract of every member of a class K then, K is factor embeddable - in particular if the members of K have one element submodels, then K is factor embeddable. [2] (A retract of a model is a submodel which is also a quotientPlanetmathPlanetmath model such that the quotient map composed with the submodel embedding is the identity map.)

0.0.2 Examples

The following are examples of factor embeddable classes:

  • The varietyMathworldPlanetmathPlanetmath of all groups (the trivial group is a one element subalgebraPlanetmathPlanetmathPlanetmath of every group)

  • The variety of all lattices (every latticeMathworldPlanetmath has one element sublattices)

  • The class of all non-trivial Boolean algebrasMathworldPlanetmath (the two element Boolean algebra is a retract of all non-trivial Boolean algebras)

The class of all Boolean algebras is an example of a class which is not factor embeddable - there is no way to embed the trivial Boolean algebra into a product of itself with any non-trivial Boolean algebras. (The trivial Boolean algebra satisfies the identity 0=1 which is not satisfied by any Boolean algebra having more than one element.)


  • 1 Peter Bruyns, Henry Rose: Varieties with cofinal sets: examples and amalgamation, Proc. Amer. Math. Soc. 111 (1991), 833-840
  • 2 Colin Naturman, Henry Rose: Ultra-universal models, Quaestiones Mathematicae, 15(2), 1992, 189-195
Title factor embeddable
Canonical name FactorEmbeddable
Date of creation 2013-03-22 19:36:55
Last modified on 2013-03-22 19:36:55
Owner Naturman (26369)
Last modified by Naturman (26369)
Numerical id 19
Author Naturman (26369)
Entry type Definition
Classification msc 03C52
Related topic JointEmbeddingProperty
Defines factor embeddable class
Defines factor embedding
Defines strong factor embedding