ultra-universal

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

ϕ

and

ψ

,

T

⊢

ϕ

∨

ψ

implies

T

⊢

ϕ

or

T

⊢

ψ

.

[

3

]

A

complete

consistent theory is always ultra-universal. More generally the set of universal sentences

Σ

of a complete consistent theory

T

is always an ultra-universal theory - a model of

T

is an ultra-universal model of

Σ

. 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

∩

A

=

U

∩

A

where

A

is the sub-lattice of universal sentences. Moreover

T

is ultra-universal iff

F

∩

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

]

References

1

Peter Bruyns, Henry Rose:

Varieties with

cofinal

sets: examples and amalgamation

, Proc. Amer. Math. Soc. 111 (1991), 833-840

2

Colin Naturman, Henry Rose:

Interior algebras: some universal algebraic aspects

, J. Korean Math. Soc. 30 (1993), No. 1, pp. 1-23

3

Colin Naturman, Henry Rose:

Ultra-universal models

, Quaestiones Mathematicae, 15(2), 1992, 189-195

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

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