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 $K$. [3]

0.0.1 Characterizations

Ultra-universal classes are precisely the non-empty elementary classes having the joint embedding property. [3]

Ultra-universal 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 ultra-universal 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 ultra-universal iff it is consistent and for all universal sentences $\varphi$ and $\psi$, $T⊢\varphi \vee \psi$ implies $T⊢\varphi$ or $T⊢\psi$. [3]

A complete consistent theory is always ultra-universal. More generally the set of universal sentences $\mathrm{\Sigma }$ of a complete consistent theory $T$ is always an ultra-universal theory - a model of $T$ is an ultra-universal model of $\mathrm{\Sigma }$. 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 $T$. [3]

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. [3]

0.0.2 Examples

• Any infinite partition lattice is ultra-universal in the variety of lattices [1]

• Any infinite symmetric group is ultra-universal in the variety of groups [3]

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

• The semigroup reduct of the monoid of functions defined on an infinite set is ultra-universal in the varierty of semigroups [3]

• The power set interior (or closure) algebra on Cantor’s discontinuum or on a denumerable co-finite topological space is ultra-universal 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 ultra-universal in the class. [3]

• 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. [3]