freely generated inductive set

freely generated inductive set

In the parent entry, we see that an

inductive set

is a set that is

closed under

the

successor

operator

. If

A

is a non-empty inductive set, then

ℕ

can be embedded in

A

.

More generally, fix a non-empty set

U

and a set

F

of finitary operations on

U

. A set

A

⊆

U

is said to be

inductive

(with respect to

F

) if

A

is closed under each

f

∈

F

. This means, for example, if

f

is a

binary operation

on

U

and if

x

,

y

∈

A

, then

f

⁢

(

x

,

y

)

∈

A

.

A

is said to be inductive over

X

if

X

⊆

A

. The

intersection

of inductive sets is clearly inductive. Given a set

X

⊆

U

, the intersection of all inductive sets over

X

is said to be the

inductive closure

of

X

. The inductive closure of

X

is written

⟨

X

⟩

. We also say that

X

generates

⟨

X

⟩

.

Another way of defining

⟨

X

⟩

is as follows: start with

X

0

:=

X

.

Next, we “inductively” define each

X

i

+

1

from

X

i

, so that

X

i

+

1

:=

X

i

∪

⋃

{

f

⁢

(

X

i

n

)

∣

f

∈

F

,

f

⁢

is

⁢

n

⁢

-ary

}

.

Finally, we set

X

¯

:=

⋃

i

=

0

∞

X

i

.

It is not hard to see that

X

¯

=

⟨

X

⟩

.

Proof.

By

definition

,

X

⊆

X

¯

. Suppose

f

∈

F

is

n

-ary, and

a

1

,

…

,

a

n

∈

X

¯

, then each

a

i

∈

X

m

⁢

(

i

)

. Take the maximum

m

of the integers

m

⁢

(

i

)

, then

a

i

∈

X

m

for each

i

. Therefore

f

⁢

(

a

1

,

…

,

a

n

)

∈

X

m

+

1

⊆

X

¯

. This shows that

X

¯

is inductive over

X

, so

⟨

X

⟩

⊆

X

¯

, since

⟨

X

⟩

is

minimal

. On the other hand, suppose

a

∈

X

¯

. We prove by

induction

that

a

∈

⟨

X

⟩

. If

a

∈

X

, this is clear. Suppose now that

X

i

⊆

⟨

X

⟩

, and

a

∈

X

i

+

1

. If

a

∈

X

i

, then we are done. Suppose now

a

∈

X

i

+

1

-

X

i

. Then there is some

n

-ary

operation

f

∈

F

, such that

a

=

f

⁢

(

a

1

,

…

,

a

n

)

, where each

a

j

∈

X

i

. So

a

j

∈

⟨

X

⟩

by

hypothesis

. Since

⟨

X

⟩

is inductive,

f

⁢

(

a

1

,

…

,

a

n

)

∈

⟨

X

⟩

, and hence

a

∈

⟨

X

⟩

as well. This shows that

X

i

+

1

⊆

A

, and consequently

X

¯

⊆

⟨

X

⟩

.
∎

The inductive set

A

is said to be

freely generated

by

X

(with respect to

F

), if the following conditions are satisfied:

1.

A

=

⟨

X

⟩

,

2.

for each

n

-ary

f

∈

F

, the

restriction

of

f

to

A

n

is

one-to-one

;

3.

for each

n

-ary

f

∈

F

,

f

⁢

(

A

n

)

∩

X

=

∅

;

4.

if

f

,

g

∈

F

are

n

,

m

-ary, then

f

⁢

(

A

n

)

∩

g

⁢

(

A

m

)

=

∅

.

For example, the set

V

¯

of

well-formed formulas

(wffs) in the classical

proposition

logic is inductive over the set of

V

propositional variables

with respect to the

logical connectives

(say,

¬

and

∨

) provided. In fact, by

unique readability

of wffs,

V

¯

is freely generated over

V

. We may readily interpret the above “freeness” conditions as follows:

1.

V

¯

is generated by

V

,

2.

for distinct wffs

p

,

q

, the wffs

¬

⁢

p

and

¬

⁢

q

are distinct; for distinct pairs

(

p

,

q

)

and

(

r

,

s

)

of wffs,

p

∨

q

and

r

∨

s

are distinct also

3.

for no wffs

p

,

q

are

¬

⁢

p

and

p

∨

q

propositional variables

4.

for wffs

p

,

q

, the wffs

¬

⁢

p

and

p

∨

q

are never the same

A

characterization

of free generation is the following:

Proposition 1

.

The following are

equivalent

:

1.

A

is freely generated by

X

(with respect to

F

)

2.

if

V

≠

∅

is a set, and

G

is a set of finitary operations on

V

such that there is a

function

ϕ

:

F

→

G

taking every

n

-ary

f

∈

F

to an

n

-ary

ϕ

⁢

(

f

)

∈

G

, then every function

h

:

X

→

B

has a unique

extension

h

¯

:

A

→

B

such that

h

¯

⁢

(

f

⁢

(

a

1

,

…

,

a

n

)

)

=

ϕ

⁢

(

f

)

⁢

(

h

¯

⁢

(

a

1

)

,

…

,

h

¯

⁢

(

a

n

)

)

,

where

f

is an

n

-ary operation in

F

, and

a

i

∈

A

.

References

1

H. Enderton:

A Mathematical

Introduction

to Logic

, Academic Press, San Diego (1972).

Title

freely generated inductive set

Canonical

name

FreelyGeneratedInductiveSet

Date of creation

2013-03-22 18:51:24

Last modified on

2013-03-22 18:51:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B99

Classification

msc 03E20

Related topic

ClosureOfSetsClosedUnderAFinitaryOperation

Defines

inductive closure

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