ultrauniversal
Let $T$ be a first order theory. A model $M$ of $T$ is said to be an ultrauniversal model of $T$ iff for every model $A$ of $T$ there exists and ultrapower of $M$ into which $A$ can be embedded. [1, 3]
If $T$ has an ultrauniversal model it is referred to as an ultrauniversal theory. The class of models of an ultrauniversal theory is called an ultrauniversal class. If $T$ is an ultrauniversal theory with elementary class $K$ and ultrauniversal model $M$ then $M$ is said to be ultrauniversal in $K$. [3]
0.0.1 Characterizations
Ultrauniversal classes are precisely the nonempty elementary classes having the joint embedding property. [3]
Ultrauniversal models can be characterized in terms of universal^{} or existential sentences:
Let $T$ be theory and let $M$ be a model of $T$. The following are equivalent^{}: [3]

1.
$M$ is an ultrauniversal model of $T$

2.
Every universal sentence holding in $M$ holds in all models of $T$

3.
Every existential sentence holding in some model of $T$ holds in $M$
A theory $T$ is ultrauniversal iff it is consistent and for all universal sentences $\varphi $ and $\psi $, $T\u22a2\varphi \vee \psi $ implies $T\u22a2\varphi $ or $T\u22a2\psi $. [3]
A complete^{} consistent theory is always ultrauniversal. More generally the set of universal sentences $\mathrm{\Sigma}$ of a complete consistent theory $T$ is always an ultrauniversal theory  a model of $T$ is an ultrauniversal model of $\mathrm{\Sigma}$. Ultrauniversal 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 deducible^{} from $T$. [3]
In terms of the LindenbaumTarski algebra for a first order language $L$, a theory $T$ in $L$ is ultrauniversal iff the filter $F$ that it generates in the LindenbaumTarski algebra is proper and can be extended to an ultrafilter^{} $U$ such that $F\cap A=U\cap A$ where $A$ is the sublattice of universal sentences. Moreover $T$ is ultrauniversal iff $F\cap A$ is a prime proper filter in $A$. Thus ultrauniversal theories correspond to prime proper filters in the bounded^{} distributive lattice^{} of universal sentences. [3]
0.0.2 Examples

•
Any infinite^{} partition lattice is ultrauniversal in the variety^{} of lattices [1]

•
Any infinite symmetric group is ultrauniversal in the variety of groups [3]

•
The monoid of functions defined on an infinite set is ultrauniversal in the variety of monoids [3]
 •

•
The power set^{} interior (or closure^{}) algebra^{} on Cantor’s discontinuum or on a denumerable cofinite topological space^{} is ultrauniversal in the variety of interior (or closure) algebras [2]

•
The product^{} of all fintely generated substructures (up to isomorphism^{}) of members of a factor embeddable universal Horn class (in particular a factor embeddable variety of algebraic structures^{}) is ultrauniversal in the class. [3]
 •
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: Interior algebras: some universal algebraic aspects, J. Korean Math. Soc. 30 (1993), No. 1, pp. 123
 3 Colin Naturman, Henry Rose: Ultrauniversal models, Quaestiones Mathematicae, 15(2), 1992, 189195
Title  ultrauniversal 
Canonical name  Ultrauniversal 
Date of creation  20130322 19:36:18 
Last modified on  20130322 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  ultrauniversal model 
Defines  ultrauniversal theory 
Defines  ultrauniversal class 