Myhill-Nerode theorem for semigroups

Myhill-Nerode theorem for semigroups

Let

S

be a

semigroup

.

X

⊆

S

is

recognizable

if it is union of classes of a

congruence

χ

such that

S

/

χ

is finite.

In the rest of the entry,

≡

X

will be the

syntactic congruence

of

X

,
and

𝒩

X

its

Nerode equivalence

.

Theorem 1

(Myhill-Nerode theorem for semigroups)

Let

S

be a semigroup and let

X

⊆

S

.
The following are

equivalent

.

1.

X

is recognizable.

2.

There exist a finite semigroup

T

and a

morphism

ϕ

:

S

→

T

such that

X

=

ϕ

-

1

⁢

(

ϕ

⁢

(

X

)

)

.

3.

The syntactic semigroup of

X

is finite.

4.

There exists an

equivalence relation

∼

on

S

such that

–

S

/

∼

is finite and

–

s

1

∼

s

2

implies

s

1

⁢

t

∼

s

2

⁢

t

.

5.

The quotient set

S

/

𝒩

X

is finite.

Theorem

1

generalizes

Myhill-Nerode theorem for languages

to subsets of

generic

, not necessarily free, semigroups.
In fact, as a

consequence

of Theorem

1

,
a

language

on a finite

alphabet

is recognizable in the sense given above
if and only if it is recognizable by a DFA.

The equivalence of points

1

,

2

and

3

is attributed to John Myhill,
while the equivalence of points

1

,

4

and

5

is attributed to Anil Nerode.

Proof of Theorem

1

.

1

⇒

2

.
Given a congruence

χ

such that

|

S

/

χ

|

<

∞

and

X

is union of classes of

χ

,
choose

T

as the quotient semigroup

S

/

χ

and

ϕ

as the

natural homomorphism

mapping

s

to

[

s

]

χ

.

2

⇒

3

.
Given a morphism of semigroups

ϕ

:

S

→

T

with

T

finite and

X

=

ϕ

-

1

⁢

(

ϕ

⁢

(

X

)

)

,
put

s

⁢

χ

⁢

t

iff

ϕ

⁢

(

s

)

=

ϕ

⁢

(

t

)

.

Since

ϕ

is a morphism,

χ

is a congruence;
moreover,

X

=

ϕ

-

1

⁢

(

ϕ

⁢

(

X

)

)

means that

X

is union of classes of

χ

-equivalence.
By the maximality property of syntactic congruence,
the number of classes of

≡

X

does not exceed the number of classes of

χ

,
which in turn does not exceed the number of elements of

T

.

3

⇒

1

.
Straightforward from

X

being union of classes of

≡

X

.

3

⇒

4

.
Straightforward from

≡

X

satisfying the second requirement.

4

⇒

5

.
Straightforward from the maximality property of Nerode equivalence.

5

⇒

3

.
Let

S

/

𝒩

X

=

{

[

ξ

1

]

𝒩

X

,

…

,

[

ξ

K

]

𝒩

X

}

.

(1)

Since

s

1

≡

X

s

2

iff

ℓ

⁢

s

1

⁢

𝒩

X

⁢

ℓ

⁢

s

2

for every

ℓ

∈

S

,
determining

[

s

]

≡

X

is the same as determining

[

ℓ

⁢

s

]

𝒩

X

as

ℓ

varies in

S

,

plus the class

[

s

]

N

X

.
(This additional class takes into account the possibility
that

s

∉

[

ℓ

⁢

s

]

𝒩

X

for any

ℓ

∈

S

,
which cannot be excluded

a priori

:
do not forget that

S

is not required to be a monoid.)

But since

s

1

⁢

𝒩

X

⁢

s

2

implies

s

1

⁢

t

⁢

𝒩

X

⁢

s

2

⁢

t

,
if

[

ℓ

1

]

𝒩

X

=

[

ℓ

2

]

𝒩

X

then

[

ℓ

1

⁢

s

]

𝒩

X

=

[

ℓ

2

⁢

s

]

𝒩

X

as well.
To determine

[

s

]

≡

X

it is thus

sufficient

to determine the

[

ξ

i

⁢

s

]

𝒩

X

for

1

≤

i

≤

K

,
plus

[

s

]

𝒩

X

.

We can thus identify the class

[

s

]

≡

X

with the

sequence

(

[

ξ

1

⁢

s

]

𝒩

X

,

…

,

[

ξ

K

⁢

s

]

𝒩

X

,

[

s

]

𝒩

X

)

.

Then the number of classes of

≡

X

cannot exceed that of

(

K

+

1

)

-ples of classes of

𝒩

X

,
which is

K

K

+

1

.

□

References

A. de Luca and S. Varricchio.

Finiteness and

Regularity

in Semigroups and

Formal Languages

.

Springer Verlag, Heidelberg 1999.

Title

Myhill-Nerode theorem for semigroups

Canonical name

MyhillNerodeTheoremForSemigroups

Date of creation

2014-05-14 17:29:14

Last modified on

2014-05-14 17:29:14

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 20M35

Classification

msc 68Q70

Defines

recognizable subset of a semigroup

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