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]

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