# ultra-universal

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 $K$. 

## 0.0.1 Characterizations

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

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 $\phi$ and $\psi$, $T\vdash\phi\vee\psi$ implies $T\vdash\phi$ or $T\vdash\psi$. 

A complete      consistent theory is always ultra-universal. More generally the set of universal sentences $\Sigma$ of a complete consistent theory $T$ is always an ultra-universal theory - a model of $T$ is an ultra-universal model of $\Sigma$. 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 deducible  from $T$. 

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 ultrafilter   $U$ such that $F\cap A=U\cap A$ where $A$ is the sub-lattice of universal sentences. Moreover $T$ is ultra-universal iff $F\cap A$ is a prime proper filter in $A$. Thus ultra-universal theories correspond to prime proper filters in the bounded    distributive lattice  of universal sentences. 

## 0.0.2 Examples

• Any infinite symmetric group is ultra-universal in the variety of groups 

• The monoid of functions defined on an infinite set is ultra-universal in the variety of monoids 

• The semigroup reduct of the monoid of functions defined on an infinite set is ultra-universal in the varierty of semigroups 

• More generally, any model in an elementary class having the property that all fintely generated substructures of the class are embeddable in it, is ultra-universal in the class. 

## References

 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