many-sorted structure

many-sorted structure

Let

L

be a

many-sorted language

and

S

the set of sorts. A

many-sorted structure

M

for

L

, or simply an

L

-

structure

consists of the following:

1.

for each sort

s

∈

S

, a non-empty set

A

s

,

2.

for each

function symbol

f

of sort type

(

s

1

,

…

,

s

n

)

:

–

if

n

>

1

, a

function

f

M

:

A

s

1

×

⋯

×

A

s

n

-

1

→

A

s

n

–

if

n

=

1

(constant symbol), an

element

f

M

∈

A

s

1

3.

for each relation symbol

r

of sort type

(

s

1

,

…

,

s

n

)

, a

relation

(or subset)

r

M

⊆

A

s

1

×

⋯

×

A

s

n

.

A

many-sorted algebra

is a many-sorted structure without any relations.

Remark

. A many-sorted structure is a special case of a more general

concept

called a

many-sorted interpretation

, which consists all of items 1-3 above, as well as the following:

4.

an element

x

M

∈

A

s

for each

variable

x

of sort

s

.

Examples

.

1.

A

left module

over a ring can be thought of as a two-sorted

algebra

(say, with sorts

{

s

1

,

s

2

}

), for there are

–

there are two non-empty sets

M

(corresponding to sort

s

1

) and

R

(corresponding to sort

s

2

), where

–

M

has the structure of an

abelian group

(equipped with three

operations

:

0

,

-

,

+

, corresponding to function symbols of sort types

(

s

1

)

,

(

s

1

,

s

1

)

, and

(

s

1

,

s

1

,

s

1

)

)

–

R

has the structure of a ring (equipped with at least four operations:

0

,

-

,

+

,

×

, corresponding to function symbols of sort types

(

s

2

)

,

(

s

2

,

s

2

)

and

(

s

2

,

s

2

,

s

2

)

for

+

and

×

, and possibly a fifth operation

1

of sort type

(

s

2

)

)

–

a function

⋅

:

R

×

M

→

M

, which corresponds to a function symbol of sort type

(

s

2

,

s

1

,

s

1

)

. Clearly,

⋅

is the scalar multiplication on the module

M

.

For a right module over a ring, one merely replaces the sort type of the last function symbol by the sort type

(

s

1

,

s

2

,

s

1

)

.

2.

A

deterministic

semiautomaton

A

=

(

S

,

Σ

,

δ

)

is a two-sorted algebra, where

–

S

and

Σ

are non-empty sets, corresponding to sorts, say,

s

1

and

s

2

,

–

δ

:

S

×

Σ

→

S

is a function corresponding to a function symbol of sort type

(

s

1

,

s

2

,

s

1

)

.

3.

A

deterministic automaton

B

=

(

S

,

Σ

,

δ

,

σ

,

F

)

is a two-sorted structure, where

–

(

S

,

Σ

,

δ

)

is a semiautomaton discussed earlier,

–

σ

is a

constant

corresponding to a nullary function symbol of sort type

(

s

1

)

,

–

F

is a

unary relation

corresponding to a relation symbol of sort type

(

s

1

)

.

Because

F

is a relation,

B

is not an algebra.

4.

A

complete sequential machine

M

=

(

S

,

Σ

,

Δ

,

δ

,

λ

)

is a three-sorted algebra, where

–

(

S

,

Σ

,

δ

)

is a semiautomaton discussed earlier,

–

Δ

is a non-empty sets, corresponding to sort, say,

s

3

,

–

λ

:

S

×

Σ

→

Δ

is a function corresponding to a function symbol of sort type

(

s

1

,

s

2

,

s

3

)

.

References

1

J. D. Monk,

Mathematical Logic

, Springer, New York (1976).

Title

many-sorted structure

Canonical

name

ManysortedStructure

Date of creation

2013-03-22 17:45:17

Last modified on

2013-03-22 17:45:17

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03B70

Classification

msc 03B10

Classification

msc 03C07

Synonym

many sorted structure

Synonym

many sorted algebra

Defines

many-sorted interpretation

Defines

many-sorted algebra

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