imaginaries

imaginaries

Given an

algebraic structure

S

to investigate, mathematicians consider

substructures

,

restrictions

of the

structure

,

quotient structures

and the like. A natural question for a mathematician to ask if he is to understand

S

is “What structures naturally live in

S

?” We can formalise this question in the following manner: Given some logic appropriate to the structure

S

, we say another structure

T

is

definable

in

S

iff there is some definable subset

T

′

of

S

n

, a

bijection

σ

:

T

′

→

T

and a definable function (respectively

relation

) on

T

′

for each

function

(resp. relation) on

T

so that

σ

is an

isomorphism

(of the relevant type for

T

).

For an example take some

infinite group

(

G

,

.

)

. Consider the centre of

G

,

Z

:=

{

x

∈

G

:

∀

y

∈

G

(

x

y

=

y

x

)

}

. Then

Z

is a

first order

definable subset of

G

, which forms a

group with

the restriction of the

multiplication

, so

(

Z

,

.

)

is a first order

definable structure

in

(

G

,

.

)

.

As another example consider the structure

(

𝐑

,

+

,

.

,

0

,

1

)

as a field. Then the structure

(

𝐑

,

<

)

is first order definable in the structure

(

𝐑

,

+

,

.

,

0

,

1

)

as for all

x

,

y

∈

𝐑

2

we have

x

≤

y

iff

∃

z

(

z

2

=

y

-

x

)

. Thus we know that

(

𝐑

,

+

,

.

,

0

,

1

)

is unstable as it has a definable order on an

infinite subset

.

Returning to the first example,

Z

is normal in

G

, so the set of (left) cosets of

Z

form a

factor group

. The domain of the factor group is the

quotient

of

G

under the

equivalence relation

x

≡

y

iff

∃

z

∈

Z

(

x

z

=

y

)

. Therefore the factor group

G

/

Z

will not (in general) be a definable structure, but would seem to be a “natural” structure. We therefore weaken our formalisation of “natural” from definable to interpretable. Here we require that a structure is isomorphic to some definable structure on

equivalence classes

of definable equivalence relations. The equivalence classes of a

∅

-definable equivalence relation are called

imaginaries

.

In

[

2

]

Poizat defined the property of

Elimination of Imaginaries

. This is

equivalent

to the following definition:

Definition 0.1

A structure

A

with at least two distinct

∅

-definable

elements

admits

elimination of imaginaries

iff for every

n

∈

N

and

∅

-definable equivalence relation

∼

on

A

n

there is a

∅

-definable function

f

:

A

n

→

A

p

(for some

p

) such that for all

x

and

y

from

A

n

we have

x

∼

y

⁢

iff

⁢

f

⁢

(

x

)

=

f

⁢

(

y

)

.

Given this property, we think of the function

f

as coding the equivalence classes of

∼

, and we call

f

⁢

(

x

)

a code for

x

/

∼

. If a structure has elimination of imaginaries then every interpretable structure is definable.

In

[

3

]

Shelah defined, for any structure

𝔄

a multi-sorted structure

𝔄

e

⁢

q

. This is done by adding a sort for every

∅

-definable equivalence relation, so that the equivalence classes are elements (and code themselves). This is a

closure operator

i.e.

𝔄

e

⁢

q

has elimination of imaginaries. See

[

1

]

chapter 4 for a good

presentation

of imaginaries and

𝔄

e

⁢

q

.
The idea of passing to

𝔄

e

⁢

q

is very useful for many purposes. Unfortunately

𝔄

e

⁢

q

has an unwieldy

language

and theory. Also this approach does not answer the question above. We would like to show that our structure has elimination of imaginaries with just a small selection of sorts added, and perhaps in a simple language. This would allow us to describe the definable structures more easily, and as we have elimination of imaginaries this would also describe the interpretable structures.

References

1

Wilfrid Hodges,

A shorter

model theory

Cambridge University Press, 1997.

2

Bruno Poizat,

Une théorie de Galois imaginaire

, Journal of Symbolic Logic,

48

(1983), pp. 1151-1170.

3

Saharon Shelah,

Classification Theory and the Number of Non-isomorphic Models

, North Hollans, Amsterdam, 1978.

Title

imaginaries

Canonical name

Imaginaries

Date of creation

2013-03-22 13:25:50

Last modified on

2013-03-22 13:25:50

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C95

Classification

msc 03C68

Related topic

CyclicCode

Defines

imaginaries

Defines

elimination of imaginaries

Defines

definable structure

Defines

interpretable structure

Defines

code

Generated on Thu Feb 8 19:56:47 2018 by

LaTeXML