monadic algebra

monadic algebra

Let

B

be a

Boolean algebra

. An

existential quantifier operator

on

B

is a

function

∃

:

B

→

B

such that

1.

∃

(

0

)

=

0

,

2.

a

≤

∃

(

a

)

, where

a

∈

B

, and

3.

∃

(

a

∧

∃

(

b

)

)

=

∃

(

a

)

∧

∃

(

b

)

, where

a

,

b

∈

B

.

A

monadic algebra

is a pair

(

B

,

∃

)

, where

B

is a Boolean algebra and

∃

is an existential quantifier operator.

There is an

obvious

connection

between an existential quantifier operator on a Boolean algebra and an

existential quantifier

in a

first order logic

:

1.

A statement

φ

⁢

(

x

)

is false iff

∃

x

⁢

φ

⁢

(

x

)

is false. For example, suppose

x

is a

real number

. Let

φ

⁢

(

x

)

be the statement

x

=

x

+

1

. Then

φ

⁢

(

x

)

is false no matter what

x

is. Likewise,

∃

φ

⁢

(

x

)

is always false too.

2.

φ

⁢

(

x

)

implies

∃

x

⁢

φ

⁢

(

x

)

; in other words, if

∃

x

⁢

φ

⁢

(

x

)

is false, then so is

φ

⁢

(

x

)

. For example, let

φ

⁢

(

x

)

be the statement

1

<

x

, where

x

∈

ℝ

. By itself,

φ

⁢

(

x

)

is neither true nor false. However

∃

x

⁢

φ

⁢

(

x

)

is always true.

3.

∃

x

⁢

(

φ

⁢

(

x

)

∧

∃

x

⁢

ψ

⁢

(

x

)

)

iff

∃

x

⁢

φ

⁢

(

x

)

∧

∃

x

⁢

ψ

⁢

(

x

)

. For example, suppose again

x

is real. Let

φ

⁢

(

x

)

be the statement

x

<

1

and

ψ

⁢

(

x

)

the statement

x

>

1

. Then both

∃

x

⁢

ψ

⁢

(

x

)

and

∃

x

⁢

φ

⁢

(

x

)

are true. It is easy to verify the

equivalence

of the two

sentences

in this example. Notice that, however,

∃

x

⁢

(

φ

⁢

(

x

)

∧

ψ

⁢

(

x

)

)

is false.

Remarks

•

One may replace condition 3. above with the following three conditions to get an

equivalent

definition

of an existential quantifier operator:

(a)

∃

(

∃

(

a

)

)

=

∃

(

a

)

(b)

∃

(

a

∨

b

)

=

∃

(

a

)

∨

∃

(

b

)

(c)

∃

(

(

∃

a

)

′

)

=

(

∃

a

)

′

From this, it is easy to see that

∃

is a

closure operator

on

B

, and that

∃

a

and

(

∃

a

)

′

are both

closed under

∃

.

•

Like the

Lindenbaum algebra

of

propositional logic

, monadic algebra is an attempt at converting first order logic into an

algebra

so that a logical question may be turned into an algebraic one. However, the existential quantifier operator in a monadic algebra corresponds to existential quantifier applied to

formulas

with only one

variable

(hence the name

monadic

). Formulas with multiple variables, such as

x

2

+

y

2

=

1

,

x

≤

y

+

z

, or

x

i

=

x

i

+

1

+

x

i

+

2

where

i

=

0

,

1

,

2

,

…

require further

generalizations

to what is known as a

polyadic algebra

. The notions of monadic and polyadic algebras were introduced by Paul Halmos.

Dual to the notion of an existential quantifier is that of a

universal quantifier

. Likewise, there is a dual of an existential quantifier operator on a Boolean algebra, a

universal quantifier operator

. Formally, a

universal quantifier operator

on a Boolean algebra

B

is a function

∀

:

B

→

B

such that

1.

∀

(

1

)

=

1

,

2.

∀

(

a

)

≤

a

, where

a

∈

B

, and

3.

∀

(

a

∨

∀

(

b

)

)

=

∀

(

a

)

∨

∀

(

b

)

, where

a

,

b

∈

B

.

Every existential quantifier operator

∃

on a Boolean algebra

B

induces a universal quantifier operator

∀

, given by

∀

(

a

)

:=

(

∃

(

a

′

)

)

′

.

Conversely

, every universal quantifier operator induces an existential quantifier by exchanging

∀

and

∃

in the definition above. This shows that the two

operations

are dual to one another.

References

1

P. Halmos, S. Givant,

Logic as Algebra

, The

Mathematical Association of America

(1998).

Title

monadic algebra

Canonical

name

MonadicAlgebra

Date of creation

2013-03-22 17:48:57

Last modified on

2013-03-22 17:48:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Classification

msc 06E25

Related topic

QuantifierAlgebra

Related topic

PolyadicAlgebra

Defines

existential quantifier operator

Defines

universal quantifier operator

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