factor embeddable
Let $K$ be a class of models (structures^{}) of a given signature^{}. Consider a (nonempty) family of structures $\{{A}_{i}:i\in I\}$ in $K$. If $j\in I$ and $f:{A}_{j}\to {\prod}_{i\in I}{A}_{i}$ is an embedding^{}, we say that $f$ is a factor embedding. [1, 2] If additionally $f$ satisfies the condition that ${\pi}_{j}\circ f$ is the identity^{} on ${A}_{i}$, where ${\pi}_{j}:{\prod}_{i\in I}{A}_{i}\to {A}_{j}$ is the $j$th projection, then $f$ is said to be a strong factor embedding. [2] $K$ is said to be a factor embeddable class iff for every (nonempty) family of structures $\{{A}_{i}:i\in I\}$ in $K$ and every $j\in I$ there is a factor embedding $f:{A}_{j}\to {\prod}_{i\in I}{A}_{i}$. [1, 2]
The definition above does not require the product^{} ${\prod}_{i\in I}{A}_{i}$ to be a member of $K$, however many interesting examples of factor embeddable classes are in fact closed under^{} 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 concepts^{} of product or embedding:
The following are equivalent^{} for a class $K$ of models [2]:

1.
$K$ is factor embeddable.

2.
For every pair of models $A,B\in K$ there exists a homomorphism^{} $f:A\to B$.
To see the above, suppose $K$ is factor embeddable and consider models $A,B\in K$. Then there exists a factor embedding from $A$ into the product $A\times B$. Composing this embedding with the projection onto $B$ gives a homomorphism $f:A\to B$. Conversely suppose such a homomorphism $f:A\to B$ exists for all $A,B\in K$ and consider a family $\{{A}_{i}:i\in I\}$ in $K$ and $j\in I$. We can define a strong factor embedding $f:{A}_{j}\to {\prod}_{i\in I}{A}_{i}$ by choosing homomorphisms ${f}_{i}:{A}_{j}\to {A}_{i}$ for all $i\in I$ with ${f}_{j}$ the identity map on ${A}_{j}$, and then for all $a\in A$ setting $f{(a)}_{i}={f}_{i}(a)$ for each $i\in I$. [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 quotient^{} 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 variety^{} of all groups (the trivial group is a one element subalgebra^{} of every group)

•
The variety of all lattices (every lattice^{} has one element sublattices)

•
The class of all nontrivial Boolean algebras^{} (the two element Boolean algebra is a retract of all nontrivial 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 nontrivial Boolean algebras. (The trivial Boolean algebra satisfies the identity $0=1$ which is not satisfied by any Boolean algebra having more than one element.)
References
 1 Peter Bruyns, Henry Rose: Varieties with cofinal sets: examples and amalgamation, Proc. Amer. Math. Soc. 111 (1991), 833840
 2 Colin Naturman, Henry Rose: Ultrauniversal models, Quaestiones Mathematicae, 15(2), 1992, 189195
Title  factor embeddable 

Canonical name  FactorEmbeddable 
Date of creation  20130322 19:36:55 
Last modified on  20130322 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 