THE MAXIMAL SUBSEMIGROUPS OF THE IDEALS OF SOME SEMIGROUPS

THE MAXIMAL SUBSEMIGROUPS OF THE IDEALS OF SOME SEMIGROUPS

OF PARTIAL INJECTIONS

Ilinka Dimitrova

Faculty of Mathematics and Natural Science South-West University "Neofit Rilski"

Blagoevgrad, 2700, Bulgaria e-mail: ilinka_dimitrova@yahoo.com

and Jörg Koppitz

^∗

Institute of Mathematics, Potsdam University Potsdam, 14469, Germany

e-mail: koppitz@rz.uni-potsdam.de

Abstract

We study the structure of the ideals of the semigroup IO

ⁿ

of all isotone (order-preserving) partial injections as well as of the semigroup IM

ⁿ

of all monotone (order-preserving or order-reversing) partial in- jections on an n-element set. The main result is the characterization of the maximal subsemigroups of the ideals of IO

ⁿ

and IM

ⁿ

.

Keywords: finite transformation semigroup, isotone and monotone partial transformations, maximal subsemigroups.

2000 Mathematics Subject Classification: 20M20.

∗

Supported by Humboldt Foundation.

1. Introduction

Let X

n

= {1, 2, . . . , n} be an n - element set ordered in the usual way. The monoid P T

n

of all partial transformations of X

n

is a very interesting object.

In this paper we will multiply transformations from the right to the left and use the corresponding notation for the right to the left composition of transformations: x(αβ) = (xα)β, for x ∈ X

n

. We say that a transformation α ∈ P T

_n

is isotone (order-preserving) if x ≤ y =⇒ xα ≤ yα for all x, y from the domain of α, antitone (order-reversing) if x ≤ y =⇒ yα ≤ xα for all x, y from the domain of α and monotone if it is isotone or antitone.

In the present paper, we study the structure of the semigroups IO

_n

of all isotone partial injections and IM

n

of all monotone partial injections of X

n

. From the definition of monotone transformations, it is clear that IO

_n

⊆ IM

n

.

Some semigroups of transformations have been studied since the sixties.

In fact, presentations of the semigroup O

n

of all isotone transformations and of the semigroup P O

n

of all isotone partial transformations (excluding the permutation in both cases) were established by A˘ızen˘stat ([1]) in 1962 and by Popova ([16]), respectively, in the same year. Some years later (1971), Howie ([14]) studied some combinatorial and algebraic properties of O

n

and, in 1992, Gomes and Howie ([13]) established some more properties of O

_n

, namely its rank and idempotent rank. In recent years it has been studied in different aspects by several authors (for example [4, 15, 17, 18]). The monoid IO

_n

of all isotone partial injections of X

_n

has been the object of study since 1997 by Fernandes in various papers ([7, 8, 9]). Some basic properties of IO

n

, in particular, a description of Green’s relations, congruences and a presentation, were obtained in [2]. Ganyushkin and Mazorchuk ([12]) studied some properties of IO

n

as describe ideals, systems of generators, maximal subsemigroups and maximal inverse subsemigroups of IO

n

.

In [10], Fernandes, Gomes and Jesus gave a presentation of both the

semigroups M

n

of all monotone transformations of X

n

and the semigroup

P M

n

of all monotone partial transformations. Dimitrova and Koppitz ([4])

considered the maximal subsemigroups of M

_n

and its ideals. Delgado and

Fernandes ([3]) have computed the abelian kernels of the semigroup IM

n

.

Fernandes, Gomes and Jesus ([11]) exhibited some properties as well as a pre-

sentation for the semigroup IM

n

. Dimitrova and Koppitz ([5]) characterized

the maximal subsemigroups of IM

_n

.

In this paper we consider the ideals of the semigroups IO

n

and IM

n

. In Section 2 we describe the maximal subsemigroups of the ideals of the semi- group IO

n

. Each of the considered ideals has exactly 2(

^nr

) − 2 maximal subsemigroups. In Section 3 we characterize the maximal subsemigroups of the ideals of the semigroup IM

n

. It happens that each of the considered ideals has exactly 2(

^nr

)

+1

− 3 maximal subsemigroups.

We will try to keep the standard notation. For every partial transfor- mation α by dom α and im α we denote the domain and the image of α, re- spectively. If α is injective, the number rank α := |dom α| = |im α| is called the rank of α. Clearly, rank αβ ≤ min{rank α, rank β} and im β = im αβ as well as dom α = dom αβ if im α = dom β. From the definition of iso- tone and antitone transformation, it follows that every element α ∈ IM

n

is uniquely determined by dom α and im α satisfying |dom α| = |im α|.

Moreover, for every A, B ⊂ X

n

of the same cardinality there exists one isotone transformation α ∈ IO

n

⊆ IM

n

and one antitone transformation β ∈ IM

n

such that dom α = dom β = A and im α = im β = B. We will denote by α

_A,B

the unique isotone element α ∈ IM

_n

for which A = dom α and B = im α, and by β

_A,B

the unique antitone element β ∈ IM

n

for which A = dom β and B = im β. The elements α

A,A

, A ∈ X

n

, exhaust all idempo- tents in IO

_n

as well as in IM

_n

. For the elements β

_A,A

, we have β

²_A,A

= α

_A,A

. In case A = B = X

n

, we will use the notations α

n

and β

_n

instead of α

Xn,Xn

and β

_Xn,Xn

.

The Green’s relations L, R, J and H on IO

n

as well as on IM

n

are characterized as follows:

αLβ ⇐⇒ im α = im β αRβ ⇐⇒ dom α = dom β αJβ ⇐⇒ rank α = rank β

H = L ∩ R.

Obviously, every H-class in IO

n

contains exactly one element and every H-

class in IM

n

\ {α ∈ IM

n

: rank α ≤ 1} contains exactly two elements. In

the set {α ∈ IM

_n

: rank α ≤ 1}, every H-class contains exactly one element.

2. Maximal subsemigroups of the ideals of IO

n

The semigroup IO

n

is the union of the J-classes J

₀

, J

₁

, . . . , J

n

, where J

r

:= {α ∈ IO

n

: rank α = r} for r = 0, . . . , n.

It is well known that the ideal I(n, r) (r = 0, . . . , n) of the semigroup IO

n

is the union of J-classes J

₀

, J

₁

, . . . , J

r

, i.e.

I(n, r) = {α ∈ IO

n

: rank α ≤ r}.

Every principal factor on IO

n

is a Rees quotient I(n, r)/I(n, r − 1) (1 ≤ r ≤ n) of which we think as J

r

∪ {0} (as it is usually convenient), where the product of two elements of J

_r

is taken to be zero if it falls into I(n, r − 1).

Let us denote by Λ

r

the collection of all subsets of X

n

of cardinality r.

The R -, L - and H - classes in J

r

have the following form:

R

A

:= {α ∈ I(n, r) : dom α = A}, A ∈ Λ

r

; L

B

:= {α ∈ I(n, r) : im α = B}, B ∈ Λ

r

; H

_A,B

:= {α

_A,B

} = R

A

∩ L

B

, A, B ∈ Λ

_r

.

Clearly, each R

A

- class (L

A

- class), A ∈ Λ

r

contains exactly one idempotent α

_A,A

. Thus if E

_r

is the set of all idempotents in the class J

_r

, then |E

_r

| =

ⁿ_r

.

Since the product αβ for all α, β ∈ J

r

belongs to the class J

r

if and only if im α = dom β, it is obvious that

Lemma 1.

1. L

B

R

A

=

( J

_r

, if A = B, 0, if A 6= B.

2. α

_A,B

α

_C,D

=

( α

A,D

, if B = C, 0, if B 6= C.

Proposition 1 [7]. hJ

r

i = I(n, r), for 0 ≤ r ≤ n − 1.

Now we begin with the description of the maximal subsemigroups of the ideals of the semigroup IO

n

.

Let us denote by Dec(Λ

_r

) the set of all decompositions (N

₁

, N

₂

) of Λ

_r

, i.e. N

₁

∪ N

₂

= Λ

r

and N

₁

∩ N

₂

= ∅ where N

₁

, N

₂

6= ∅.

Definition 1. Let (N

1

, N

2

) ∈ Dec(Λ

r

) (r = 1, . . . , n − 1). Then we put S

_(N1,N2)

:= I(n, r − 1) ∪ {α

_A,B

: A ∈ N

₁

or B ∈ N

₂

}.

The maximal subsemigroups of the ideal I(n, n) = IO

_n

were described by Ganyushkin and Mazorchuk:

Theorem 1 [12]. A subsemigroup S of IO

n

is maximal if and only if S = I(n, n − 1) or S = {α

n

} ∪ S

_(N1,N2)

, where (N

₁

, N

₂

) ∈ Dec(Λ

_n−1

).

In the following, we will consider the maximal subsemigroups of the ideals I(n, r) for r = 1, . . . , n − 1.

Lemma 2. Every maximal subsemigroup in I(n, r) contains the ideal I(n, r − 1).

P roof. Let S be a maximal subsemigroup of I(n, r). Assume that J

r

⊆ S, then according to Proposition 1 it follows that I(n, r) = hJ

r

i ⊆ S, i.e.

S = I(n, r), a contradiction. Thus J

r

* S. Then S ∪ I(n, r − 1) is a proper subsemigroup of I(n, r) since I(n, r−1) is an ideal, and hence S∪I(n, r−1) = S by maximality of S. This implies I(n, r−1) ⊆ S.

Theorem 2. Let 1 ≤ r ≤ n−1. Then a subsemigroup S of I(n, r) is maximal if and only if there is an element (N

1

, N

2

) ∈ Dec(Λ

r

) with S = S

_(N1,N2)

.

P roof. Let S = S

_(N1,N2)

for some (N

₁

, N

₂

) ∈ Dec(Λ

_r

). Then S = I(n, r − 1) ∪ {α

A,B

: A ∈ N

₁

or B ∈ N

₂

}.

Therefore, if α

_A,B

∈ S then A ∈ N /

2

and B ∈ N

₁

, and thus α

_B,A

∈ S.

From Lemma 1 it follows that S is a semigroup. Really, let α

A,B

, α

C,D

∈ S, i.e. A, C ∈ N

₁

or B, D ∈ N

₂

or A ∈ N

₁

, D ∈ N

₂

. Then we have α

A,B

α

C,D

= α

A,D

∈ S for B = C and α

A,B

α

C,D

= 0 ∈ I(n, r − 1) ⊆ S for B 6= C.

Now we will show that S is maximal. Let α

_C,D

∈ I(n, r) \ S, i.e. C / ∈ N

1

and D / ∈ N

2

. Then D ∈ N

1

, since N

1

∪ N

2

= Λ

r

and so α

D,P

∈ S for all P ∈ Λ

r

and thus R

D

= {α

D,P

: P ∈ Λ

r

} ⊆ S. Moreover, we have α

_C,P

= α

_C,D

α

_D,P

, for all P ∈ Λ

_r

, by Lemma 1. Thus we obtain the R-class R

C

= {α

C,P

: P ∈ Λ

r

} ⊆ hS ∪ {α

C,D

}i. Moreover, C ∈ N

₂

and so L

C

= {α

P,C

: P ∈ Λ

r

} ⊆ S. Using Lemma 1, we have L

C

R

C

= J

r

⊆ hS ∪ {α

C,D

}i.

Thus, we obtain that hS ∪ {α

C,D

}i = I(n, r). Therefore, S is a maximal subsemigroup of the ideal I(n, r).

For the converse part let S be a maximal subsemigroup of the ideal I(n, r). From Lemma 2, we have that I(n, r − 1) ⊆ S. Then S = I(n, r − 1)

∪ T , where T ⊆ J

r

.

Let α

A,B

∈ S. Then hS∪{α /

A,B

}i = I(n, r). Let now P, Q ∈ Λ

r

. Suppose that α

_P,Q

∈ S. Then α /

P,Q

∈ hS ∪ {α

A,B

}i and α

P,Q

= α

_P,A

α

_A,B

α

_B,Q

. Moreover, α

P,A

= α

P,A

α

A,B

α

B,A

and α

B,Q

= α

B,A

α

A,B

α

B,Q

. This shows that we need α

P,A

and α

B,Q

to generate α

P,A

and α

B,Q

, respectively, with elements of S ∪ {α

_A,B

}. Hence α

P,A

, α

_B,Q

∈ S.

Assume that α

Q,P

∈ S. Then α /

Q,P

= α

Q,A

α

A,B

α

B,P

and by the same arguments, we obtain that α

_Q,A

, α

_B,P

∈ S.

Further, from α

Q,P

= α

Q,A

α

A,P

it follows that α

A,P

∈ S. But α /

P,Q

∈ / hS ∪ {α

A,P

}i since α

P,Q

= α

P,A

α

A,P

α

P,Q

. This contradicts the maximality of S and thus α

Q,P

∈ S. Hence if α

P,Q

∈ S then α /

Q,P

∈ S for any P, Q ∈ Λ

r

. Therefore, for N

₁

= {B : α

A,B

∈ S} and N /

2

= {A : α

A,B

∈ S} we have that / S = S

_(N1,N2)

.

There are exactly 2(

^nr

) − 2 maximal subsemigroups of the ideal I(n, r), for r = 1, . . . , n − 1 and 2

ⁿ

− 1 maximal subsemigroups of I(n, n).

3. Maximal subsemigroups of the ideals of IM

n

The semigroup IM

n

is the union of the J-classes J

₀

, J

₁

, . . . , J

n

, where

J

_r

:= {α ∈ IM

_n

: rank α = r} for r = 0, . . . , n.

It is well known that the ideal I(n, r) (r = 0, . . . , n) of the semigroup IM

n

is the union of J-classes J

₀

, J

₁

, . . . , J

r

, i.e.

I(n, r) = {α ∈ IM

n

: rank α ≤ r}.

Every principal factor on IM

n

is a Rees quotient I(n, r)/I(n, r − 1) (1 ≤ r ≤ n) of which we think as J

_r

∪ {0}, where the product of two elements of J

r

is taken to be zero if it falls into I(n, r − 1).

The R -, L - and H - classes in J

r

have the following form:

R

A

:= {α ∈ I(n, r) : dom α = A}, A ∈ Λ

r

; L

B

:= {α ∈ I(n, r) : im α = B}, B ∈ Λ

r

; H

_A,B

:= {α

_A,B

, β

_A,B

} = R

A

∩ L

B

, A, B ∈ Λ

_r

.

The L-class, R-class and H-class, respectively, containing the element α ∈ IM

n

will be denoted by L

α

, R

α

, and H

α

, respectively.

Since the product αβ for all α, β ∈ J

r

belongs to the class J

r

if and only if im α = dom β, it is easy to show that

Lemma 3.

1. L

B

R

A

=

( J

_r

, if A = B, 0, if A 6= B.

2. H

A,B

H

C,D

=

( H

A,D

, if B = C, 0, if B 6= C.

Let U be a subset of the semigroup IM

n

. We denote by U

ⁱ

(respectively U

^a

) the set of all isotone (respectively antitone) transformations in the set U . An immediate but important property is that the product of two isotone transformations or two antitone transformations is an isotone, and the prod- uct of an isotone transformation with an antitone transformation, or vice versa, is an antitone one.

Proposition 2. J

r

⊆ hJ

_r^a

i and J

r

⊆ hJ

_rⁱ

∪ {β

_A,B

}i, for all A, B ∈ Λ

r

.

P roof. Let A, B ∈ Λ

r

. Then for all C ∈ Λ

r

, we have α

A,B

= β

_A,C

β

_C,B

. Therefore, J

r

⊆ hJ

_r^a

i.

From L

ⁱ_A

β

_A,B

= L

^a_B

and L

^a_B

R

_Bⁱ

= J

_r^a

, we have J

r

⊆ hJ

_rⁱ

∪ {β

_A,B

}i.

Proposition 3. hJ

r

i = I(n, r), for 0 ≤ r ≤ n − 1.

P roof. Clearly hJ

₀

i = I(n, 0). In [5], it was shown that J

_r−1ⁱ

⊆ J

_rⁱ

J

_rⁱ

and J

_r−1^a

⊆ J

_r−1ⁱ

J

_r^a

J

_r−1ⁱ

for 1 ≤ r ≤ n − 1. Since I(n, r) = J

₀

∪ J

1

∪ · · · ∪ J

r

, we have hJ

r

i = I(n, r).

From Proposition 2 and Proposition 3 we have

Corollary 1. Let 1 ≤ r ≤ n − 1. Then hJ

_r^a

i = hJ

_rⁱ

∪ {β

_A,B

}i = I(n, r), for all A, B ∈ Λ

r

.

Now we begin with the description of the maximal subsemigroups of the ideals of the semigroup IM

n

.

Clearly, the ideal I(n, 1) of IM

n

coincides with the ideal I(n, 1) of IO

n

. Thus the maximal subsemigroups of this ideal are characterized in Theorem 2 and there are exactly 2

ⁿ

− 2 such semigroups.

Now we will consider the maximal subsemigroups of the ideals I(n, r) for r = 2, . . . , n − 1.

Lemma 4. Every maximal subsemigroup in I(n, r) contains the ideal I(n, r − 1).

The proof is similar as that in Lemma 2.

Theorem 3 Let 2 ≤ r ≤ n−1. Then a subsemigroup S of I(n, r) is maximal if and only if it belongs to one of the following three types:

(1) S

⁽¹⁾

:= I(n, r − 1) ∪ J

_rⁱ

;

(2) S

_(N⁽²⁾_1,N2)

:= S{H

α

: α ∈ S

_(N1,N2)

}, f or (N

1

, N

2

) ∈ Dec(Λ

r

);

(3) S

_(N⁽³⁾_1,N2)

:= I(n, r − 1) ∪ {α

A,B

: A, B ∈ N

1

or A, B ∈ N

2

} ∪

∪ {β

_A,B

: A ∈ N

₁

, B ∈ N

₂

or A ∈ N

₂

, B ∈ N

₁

} f or (N

1

, N

₂

) ∈ Dec(Λ

r

).

P roof.

(1) It is obvious that S

⁽¹⁾

= I(n, r − 1) ∪ J

_rⁱ

is a semigroup, since I(n, r − 1) is an ideal and (J

_rⁱ

)

²

⊆ I

ⁱ

(n, r) ⊆ I(n, r −1)∪J

_rⁱ

. From Proposition 2, we have that J

_r

⊆ hJ

_rⁱ

∪ {β

_A,B

}i for all β

_A,B

∈ J

_r^a

. Since I(n, r) \ S

⁽¹⁾

= J

_r^a

, we obtain hS

⁽¹⁾

∪ {β

_A,B

}i = I(n, r) for all β

_A,B

∈ J

_r^a

. Therefore, S

⁽¹⁾

is maximal in I(n, r).

(2) Let S = S

⁽²⁾_(N1,N2)

for some (N

1

, N

2

) ∈ Dec(Λ

r

). Then S = I(n, r − 1) ∪ {H

A,B

: A ∈ N

1

or B ∈ N

₂

}.

From Lemma 3 it follows that S is a semigroup. Really, let H

_A,B

, H

_C,D

⊆ S, i.e. A, C ∈ N

₁

or B, D ∈ N

₂

or A ∈ N

₁

, D ∈ N

₂

. Then we have H

A,B

H

C,D

= H

A,D

⊆ S for B = C and H

A,B

H

C,D

⊆ I(n, r − 1) ⊆ S for B 6= C.

Now we will show that S is maximal. Let H

_C,D

= {α

C,D

, β

_C,D

} ⊆ I(n, r) \ S, i.e. C / ∈ N

₁

and D / ∈ N

₂

. Then D ∈ N

₁

, since N

₁

∪ N

₂

= Λ

r

and so H

D,P

∈ S for all P ∈ Λ

r

and thus R

D

= S

P ∈Λr

H

D,P

⊆ S.

Moreover, we have

H

C,P

= H

C,D

H

D,P

, for P ∈ Λ

r

, by Lemma 3. Thus we obtain the R-class R

C

= S

P ∈Λ^r

H

C,P

⊆ hS ∪ H

C,D

i. Moreover, C ∈ N

2

and so L

C

= S

P ∈Λr

H

P,C

⊆ S. Using Lemma 3, we have L

_C

R

_C

= J

_r

⊆ hS ∪ H

C,D

i. Since α

C,D

= β

_C,D

β

_D,D

and β

_C,D

= α

C,D

β

_D,D

, where β

_D,D

∈ R

D

⊆ S, we obtain that hS∪{α

C,D

}i = I(n, r) and hS ∪ {β

_C,D

}i = I(n, r). Therefore, S is a maximal subsemi- group of the ideal I(n, r).

(3) Let S = S

_(N⁽³⁾_1,N2)

for some (N

1

, N

2

) ∈ Dec(Λ

r

). From Lemma 3, it follows that S is a semigroup. We will show that S is maximal. Let

V := I(n, r) \ S = {β

_A,B

: A, B ∈ N

1

or A, B ∈ N

2

} ∪

∪ {α

A,B

: A ∈ N

₁

, B ∈ N

₂

or A ∈ N

₂

, B ∈ N

₁

}

and let γ ∈ V . Then for the transformation γ we have four possibilities:

Let γ ∈ {β

_A,B

: A, B ∈ N

₁

}. Then α

C,A

∈ S (since A ∈ N

1

) and so α

C,A

β

_A,B

= β

_C,B

∈ hS ∪ {γ}i for all C ∈ N

1

. Also, we have β

_C,A

∈ S and thus β

_C,A

β

_A,B

= α

C,B

∈ hS ∪ {γ}i for all C ∈ N

2

. Since α

C,B

∈ S for all C ∈ N

₁

and β

_C,B

∈ S for all C ∈ N

2

, we obtain L

B

= S

C∈Λ^r

H

C,B

⊆ hS ∪ {γ}i. Further, β

_B,B

∈ L

B

and β

_B,B

β

_B,D

= α

B,D

for all D ∈ N

₂

as well as β

_B,B

α

_B,D

= β

_B,D

for all D ∈ N

₁

. Thus since α

_B,D

∈ S for all D ∈ N

1

and β

_B,D

∈ S for all D ∈ N

₂

, we obtain R

B

= S

D∈Λ^r

H

B,D

⊆ hS ∪ {γ}i.

From Lemma 3, we have L

B

R

B

= J

r

and therefore hS ∪ {γ}i = I(n, r).

– For γ ∈ {β

_A,B

: A, B ∈ N

₂

}, the proof is similar.

– Let γ ∈ {α

A,B

: A ∈ N

1

, B ∈ N

2

}. Then α

C,A

∈ S (since A ∈ N

1

) and so α

C,A

α

A,B

= α

C,B

∈ hS ∪ {γ}i for all C ∈ N

1

. Also, we have β

_C,A

∈ S and thus β

_C,A

α

_A,B

= β

_C,B

∈ hS ∪ {γ}i for all C ∈ N

2

. Since α

C,B

∈ S for all C ∈ N

2

and β

_C,B

∈ S for all C ∈ N

1

, we obtain L

B

= S

C∈Λr

H

C,B

⊆ hS ∪ {γ}i. Further, β

B,B

∈ L

B

and β

_B,B

α

B,D

= β

_B,D

for all D ∈ N

₂

as well as β

_B,B

β

_B,D

= α

_B,D

for all D ∈ N

₁

. Thus since α

_B,D

∈ S for all D ∈ N

2

and β

_B,D

∈ S for all D ∈ N

1

, we obtain R

B

= S

l∈Λ^r

H

B,D

⊆ hS ∪ {γ}i. From Lemma 3, we have L

B

R

B

= J

r

and therefore hS ∪ {γ}i = I(n, r).

– For γ ∈ {α

_A,B

: A ∈ N

2

, B ∈ N

1

}, the proof is similar.

Altogether, this shows that S is maximal.

For the converse part let S be a maximal subsemigroup of the ideal I(n, r). From Lemma 4, we have that I(n, r − 1) ⊆ S. Then S = I(n, r − 1)

∪ T , where T ⊆ J

r

. We consider two cases for the set T . 1. Let T = J

_rⁱ

. Then S = I(n, r − 1) ∪ J

_rⁱ

= S

⁽¹⁾

.

2. Let now T 6= J

_rⁱ

. Assume that J

_rⁱ

⊆ T . Then T = J

_rⁱ

∪ T

⁰

where

∅ 6= T

⁰

⊆ J

_r^a

. From Corollary 1, we have S = I(n, r), a contradiction.

Thus J

_rⁱ

* T . We also have that J

_r^a

* T since hJ

_r^a

i = I(n, r).

Admit that H

A,B

⊆ S or H

A,B

∩ S = ∅, for all A, B ∈ Λ

r

. Assume that

S

ⁱ

= S ∩ I

ⁱ

(n, r) is not a maximal subsemigroup of I

ⁱ

(n, r). Then there is an

isotone transformation α

_A,B

∈ I(n, r) \ S such that hS

ⁱ

∪ {α

A,B

}i is a proper

subset of I

ⁱ

(n, r). Therefore, there exists an α

C,D

∈ I(n, r) \ S such that

α

C,D

∈ hS /

ⁱ

∪ {α

A,B

}i. But hS ∪ {α

A,B

}i = I(n, r) since S is maximal and

α

C,D

= β

_C,A

α

A,B

β

_B,D

. Moreover, β

_C,A

= β

_C,A

α

A,B

α

B,A

= α

C,A

α

A,B

β

_B,A

and β

_B,D

= β

_B,A

α

_A,B

α

_B,D

= α

_B,A

α

_A,B

β

_B,D

. This shows that we need

β

_C,A

or α

C,A

and β

_B,D

or α

B,D

to generate β

_C,A

and β

_B,D

, respectively, with elements of S ∪ {α

A,B

}. This implies that β

_C,A

, α

C,A

, β

_B,D

, α

B,D

∈ S, since we assume that H

A,B

⊆ S or H

A,B

∩ S = ∅, for all A, B ∈ Λ

r

. Hence α

C,D

= α

C,A

α

A,B

α

B,D

∈ hS

ⁱ

∪ {α

A,B

}i, a contradiction. Therefore, we obtain that S

ⁱ

is maximal in I

ⁱ

(n, r). Since all maximal subsemigroups of the ideal I

ⁱ

(n, r) are of type S

_(N1,N2)

we have S = ∪{H

_α

: α ∈ S

ⁱ

} = S

_(N⁽²⁾_1,N2)

, for some (N

₁

, N

₂

) ∈ Dec(Λ

r

).

Now, admit that |H

A,B

∩ S| = 1, for some A, B ∈ Λ

r

. Suppose that α

_A,B

∈ S and β /

_A,B

∈ S. Then from α

A,B

= β

_A,B

β

_B,B

and α

_A,B

= β

_A,A

β

_A,B

, it follows that β

_A,A

, β

_B,B

∈ S. Moreover, from β /

_A,B

α

B,A

= β

_A,A

∈ S, we get α /

B,A

∈ S. Assume that β /

_B,A

∈ S. Then β /

_B,A

∈ hS ∪ {α

B,A

}i, because of the maximality of S, and since β

_B,A

= β

_B,B

α

_B,A

α

_A,A

= α

B,B

α

B,A

β

_A,A

, we obtain β

_A,A

∈ S or β

_B,B

∈ S, a contradiction, and thus β

_B,A

∈ S.

Further, let P, Q ∈ Λ

r

. Suppose that α

P,Q

∈ S. Then from α /

P,Q

= α

_P,A

β

_A,B

β

_B,Q

, it follows that if α

_P,A

∈ S then β

_B,Q

∈ S and vice versa. / Also from α

P,Q

= β

_P,A

β

_A,B

α

B,Q

, it follows that if β

_P,A

∈ S then α

B,Q

∈ S / and vice versa. Moreover, α

P,Q

∈ hS ∪ {α

A,B

}i since S is maximal. Hence α

P,Q

= α

P,A

α

A,B

α

B,Q

= β

_P,A

α

A,B

β

_B,Q

. Therefore, we have α

P,A

, α

B,Q

∈ S and β

_P,A

, β

_B,Q

∈ S or vice versa. /

Assume that β

_P,Q

∈ S. Then β /

_P,Q

∈ hS ∪ {α

A,B

}i and so β

_P,Q

= α

P,A

α

A,B

β

_B,Q

= β

_P,A

α

A,B

α

B,Q

. But we obtain already that if α

P,A

, α

B,Q

∈ S then β

_P,A

, β

_B,Q

∈ S or vice versa. Therefore, β /

_P,Q

∈ hS ∪ {α /

A,B

}i. This contradicts the maximality of S and thus β

_P,Q

∈ S.

Further, from α

_P,Q

= β

_P,Q

β

_Q,Q

and α

_P,Q

= β

_P,P

β

_P,Q

, it follows that β

_P,P

, β

_Q,Q

∈ S. Moreover, from β /

_P,Q

α

Q,P

= β

_P,P

∈ S, we get α /

Q,P

∈ / S. Assume that β

_Q,P

∈ S. Then β /

_Q,P

∈ hS ∪ {α

Q,P

}i, because of the maximality of S, and since β

_Q,P

= β

_Q,Q

α

_Q,P

α

_P,P

= α

_Q,Q

α

_Q,P

β

_P,P

, we obtain β

_P,P

∈ S or β

_Q,Q

∈ S, a contradiction, and thus β

_Q,P

∈ S.

Analogously, if β

_P,Q

∈ S we have that β /

_Q,P

∈ S and α /

P,Q

, α

Q,P

∈ S.

Suppose that α

P,Q

∈ S for some P, Q ∈ Λ

r

. Then β

_P,Q

∈ S. Otherwise, / from α

A,B

= α

A,P

β

_P,Q

β

_Q,B

∈ S it follows /

i) α

_A,P

∈ S and β /

_Q,B

∈ S, i.e. β

_A,P

∈ S and β

_Q,B

∈ S;

ii) α

A,P

∈ S and β

_Q,B

∈ S, i.e. α /

A,P

∈ S and α

Q,B

∈ S;

iii) α

A,P

∈ S and β /

_Q,B

∈ S, i.e. β /

_A,P

∈ S and α

Q,B

∈ S.

Then α

A,B

= β

_A,P

α

P,Q

β

_Q,B

= α

A,P

α

P,Q

α

Q,B

= β

_A,P

β

_P,Q

α

Q,B

, which con- tradicts that α

A,B

∈ S. /

The proof when α

_A,B

∈ S and β

_A,B

∈ S is similar. / Finally, we obtain that

(1) α

P,Q

∈ S ⇐⇒ β

_P,Q

∈ S /

for P, Q ∈ Λ

r

.

Let ρ

_r

:= {(P, Q) : α

P,Q

∈ S}. Obviously, ρ

_r

is an equivalence relation on Λ

_r

with Λ

_r

/ρ

_r

= {N

₁

, N

₂

, . . . , N

_m

} (m ≥ 2). Indeed, ρ

_r

is reflexive since E

r

⊆ S, symmetric because of the previous considerations and transitive since α

P,Q

α

Q,R

= α

P,R

∈ S for α

P,Q

, α

Q,R

∈ S. Moreover, m ≥ 2 becomes clear by J

_rⁱ

* T . Assume that the decomposition contains more than two elements, i.e. m > 2. Then there are N

₁

, N

₂

, N

₃

in our decomposition such that A ∈ N

₁

, B ∈ N

₂

and C ∈ N

₃

. Thus α

A,B

= β

_A,C

β

_C,B

∈ S, a con- tradiction. Therefore, Λ

_r

/ρ

_r

= {N

₁

, N

₂

} and S = S

_(N⁽³⁾_1,N2)

, because of (1).

There are exactly 2(

^nr

) − 2 maximal subsemigroups of the ideal I

ⁱ

(n, r) and exactly 2(

n

r

) −2 maximal subsemigroups of type (3). Taking I(n, r − 1) ∪ J

_rⁱ

into account, we get 2(

^nr

)

+1

− 3 maximal subsemigroups of the ideal I(n, r), for r = 2, . . . , n − 1.

Finally, we characterize the maximal subsemigroups of the ideal I(n, n) = IM

n

.

For A ∈ Λ

_n−1

we put A := {n + 1 − i : i ∈ A} and for N ⊆ P(X

n

) we set N := {A : A ∈ N }. Then we have

(2)

β

_A,A

α

_A,B

= β

_n

α

_A,B

= β

_A,B

,

β

_A,A

β

_A,B

= β

_n

β

_A,B

= α

A,B

,

α

B,A

β

_A,A

= α

B,A

β

_n

= β

_B,A

,

β

_B,A

β

_A,A

= β

_B,A

β

_n

= α

_B,A

.

Theorem 4. A subsemigroup S of IM

n

is maximal if and only if it belongs to one of the following three types:

(1) T := I(n, n − 1) ∪ {α

n

};

(2) T

_(N1,N2)

:= J

_n

∪ {H

α

: α ∈ S

_(N1,N2)

}, f or (N

1

, N

₂

) ∈ Dec(Λ

_n−1

) with N

₁

= N

₁

and N

₂

= N

₂

;

(3) T

_{(N,N )}

:= J

n

∪ I(n, n − 2) ∪ {α

A,B

: A, B ∈ N or A, B ∈ N }

∪ {β

_A,B

: A ∈ N, B ∈ N or A ∈ N , B ∈ N } f or (N, N ) ∈ Dec(Λ

_n−1

).

P roof. It is clear that T is a maximal subsemigroup of IM

_n

. Further, we put

Inv := {β

_A,A

: A ∈ Λ

_n−1

}.

Let (N

₁

, N

₂

) ∈ Dec(Λ

_n−1

) be a decomposition with the required proper- ties. Since Inv ⊆ T

_(N1,N2)

and by (2) it is easy to verify that T

_(N1,N2)

is a subsemigroup of IM

_n

. Since T

_(N1,N2)

\ J

n

is a maximal subsemigroup of I(n, n − 1) by Theorem 3 and J

n

⊆ T

_(N1,N2)

, it follows that T

_(N1,N2)

is a maximal subsemigroup of IM

n

. Analogously, one can show that T

_{(N,N )}

is a maximal subsemigroup of IM

_n

.

For the converse part, let S be maximal in IM

n

. Admit that J

n

* S.

Then it is easy to see that S = T . Now suppose that J

_n

⊆ S. Assume that Inv * S. Then there is an A ∈ Λ

n−1

with β

_A,A

∈ S. Since S is maximal, we / have IM

n

= hS ∪{β

_A,A

}i = S ∪{β

_A,A

} by (2). Thus S = IM

n

\{β

_A,A

}. But β

_A,A

= α

A,B

β

_B,A

for some B ∈ Λ

_n−1

with B 6= A. Since α

A,B

, β

_B,A

∈ S, we have S = IM

_n

, a contradiction. Hence Inv ⊆ S. Let S

_n−1

:= S ∩I(n, n−1).

Assume that S

_n−1

is not a maximal subsemigroup of I(n, n − 1). Clearly, S

_n−1

6= I(n, n − 1). Let γ ∈ I(n, n − 1) \ S

n−1

. Then for all δ ∈ I(n, n − 1), we have δ ∈ hS ∪ {γ}i = hS

_n−1

∪ {γ}i ∪ J

n

by (2) and since Inv ⊆ S. This shows that δ ∈ hS

_n−1

∪ {γ}i and thus hS

_n−1

∪ {γ}i = I(n, n − 1). Conse- quently, S

_n−1

is a maximal subsemigroup of I(n, n−1). Using Theorem 3 we choose all decompositions (N

₁

, N

₂

) ∈ Dec(Λ

_n−1

) such that Inv ⊆ S

_(N⁽²⁾

1,N2)

and Inv ⊆ S

⁽³⁾_(N

1,N2)

, respectively. In this way we obtain the semigroups

T

_(N1,N2)

and T

_{(N,N )}

.

It is straightforward to calculate that there are exactly 2

ⁿ⁺¹²

− 1 maximal subsemigroups of IM

n

if n is odd and exactly

³

2

(2

ⁿ

2

) − 1 maximal subsemi- groups of IM

_n

if n is even.

Acknowledgments

The authors would like to thank Professor Vítor H. Fernandes for his helpful suggestions and remarks in preparing this paper.

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Received 20 April 2009

Revised 10 November 2009