Munn tree

Munn tree

Let

X

be a

finite

set, and

(

X

∐

X

-

1

)

∗

the

free monoid with involution

on

X

. It is well known that the elements of

(

X

∐

X

-

1

)

∗

can be viewed as words on the

alphabet

(

X

∐

X

-

1

)

, i.e. as elements of the free monod on

(

X

∐

X

-

1

)

.

The

Munn tree

of the word

w

∈

(

X

∐

X

-

1

)

∗

is the

X

-

inverse

word graph

MT

⁢

(

w

)

(or

MT

X

⁢

(

w

)

if

X

needs to be specified) with vertex and edge set respectively

V

⁢

(

MT

⁢

(

w

)

)

=

red

⁢

(

pref

⁢

(

w

)

)

=

{

red

⁢

(

v

)

|

v

∈

pref

⁢

(

w

)

}

,

E

(

MT

(

w

)

)

=

{

(

v

,

x

,

red

(

v

x

)

)

∈

V

(

MT

(

w

)

)

×

(

X

∐

X

-

1

)

×

V

(

MT

(

w

)

)

}

.

The

concept

of Munn tree was created to investigate the

structure

of the

free inverse monoid

. The main result about it says that it “recognize” whether or not two different word in

(

X

∐

X

-

1

)

∗

belong to the same

ρ

X

-class, where

ρ

X

is the Wagner congruence on

X

. We recall that if

w

∈

(

X

∐

X

-

1

)

∗

[resp.

w

∈

(

X

∐

X

-

1

)

+

], then

[

w

]

ρ

X

∈

FIM

⁢

(

X

)

[resp.

[

w

]

ρ

X

∈

FIS

⁢

(

X

)

].

Theorem 1

(Munn)

Let

v

,

w

∈

(

X

∐

X

-

1

)

∗

(or

v

,

w

∈

(

X

∐

X

-

1

)

+

). Then

[

v

]

ρ

X

=

[

w

]

ρ

X

if and only if

MT

⁢

(

v

)

=

MT

⁢

(

w

)

As an immediate

corollary

of this result we obtain that the

word problem

in the free inverse monoid (and in the free inverse semigroup) is

decidable

. In fact, we can effectively build the Munn tree of an arbitrary word in

(

X

∐

X

-

1

)

∗

, and this suffice to prove wheter or not two words belong to the same

ρ

X

-class.

The Munn tree reveals also some

property

of the

ℛ

-classes of elements of the free inverse monoid, where

ℛ

is the right

Green relation

. In fact, the following result says that “essentially” the Munn tree of

w

∈

(

X

∐

X

-

1

)

∗

is the Schützenberger graph of the

ℛ

-class of

[

w

]

ρ

X

.

Theorem 2

Let

w

∈

(

X

∐

X

-

1

)

∗

. There exists an

isomorphism

(in the category of

X

-inverse word graphs)

Φ

:

MT

⁢

(

w

)

→

S

⁢

Γ

⁢

(

X

;

∅

;

[

w

]

ρ

X

)

between the Munn tree

MT

⁢

(

w

)

and the Schützenberger graph

S

⁢

Γ

⁢

(

X

;

∅

;

[

w

]

ρ

X

)

given by

Φ

V

⁢

(

v

)

=

[

v

]

ρ

X

,

∀

v

∈

V

⁢

(

MT

⁢

(

w

)

)

=

red

⁢

(

pref

⁢

(

w

)

)

,

Φ

E

⁢

(

(

v

,

x

,

red

⁢

(

v

⁢

x

)

)

)

=

(

[

v

]

ρ

X

,

x

,

[

v

⁢

x

]

ρ

X

)

,

∀

(

v

,

x

,

red

⁢

(

v

⁢

x

)

)

∈

E

⁢

(

MT

⁢

(

w

)

)

.

References

1

W.D. Munn,

Free inverse semigroups

, Proc. London Math. Soc. 30 (1974) 385-404.

2

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

3

J.B. Stephen,

Presentation

of inverse monoids

, J. Pure Appl.

Algebra

63 (1990) 81-112.

Title

Munn tree

Canonical

name

MunnTree

Date of creation

2013-03-22 16:11:59

Last modified on

2013-03-22 16:11:59

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

20

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M05

Classification

msc 20M18

Related topic

SchutzenbergerGraph

Defines

Munn tree

Generated on Fri Feb 9 19:33:27 2018 by

LaTeXML