A presentation of the dual symmetric inverse monoid

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A presentation of the dual symmetric inverse

monoid

David Easdown

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

de @ maths.usyd.edu.au

James East

Department of Mathematics, La Trobe University, Victoria 3083, Australia

james.east @ latrobe.edu.au

D. G. FitzGerald

School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart 7250, Australia

d.fitzgerald @ utas.edu.au

July 24, 2007

Abstract

The dual symmetric inverse monoidI∗

n is the inverse monoid of all

isomorphisms between quotients of ann-set. We give a monoid presenta-tion ofI∗

n and, along the way, establish criteria for a monoid to be inverse

when it is generated by completely regular elements.

1 Introduction

Inverse monoids model the partial or local symmetries of structures, generalizing the total symmetries modelled by groups. Key examples are the symmetric inverse monoid I_X _{on a set} _X _{(consisting of all bijections between subsets}

of X), and the dual symmetric inverse monoid I∗

X on X (consisting of all

isomorphisms between subobjects ofX in the category Setopp), each with an appropriate multiplication. They share the property that every inverse monoid may be faithfully represented in some I_X _{and some} I∗

X. The monoid IX∗

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the blocks of a “domain” equivalence (or partition) onX bijectively to blocks of a “range” equivalence. These objects may also be regarded as special binary relations on X called biequivalences. The appropriate multiplication involves the join of equivalences—details are found in [2], and an alternative description in [4, pp. 122–124].

1.1 Finite dual symmetric inverse monoids

In this paper we focus on finiteX, and writen={1, . . . n}andI∗

n =In∗. In a

graphical representation described in [5], the elements ofI∗

n are thought of as

graphs on a vertex set {1, . . . , n} ∪ {1′_{, . . . , n}′_} _{(consisting of two copies of} _n₎

such that each connected component has at least one dashed and one undashed vertex. This representation is not unique—two graphs are regarded as equivalent if they have the same connected components—but it facilitates visualization and is intimately connected to the combinatorial structure. Conventionally, we draw the graph of an element of I∗

n such that the vertices 1, . . . , n are in a

horizontal row (increasing from left to right), with vertices 1′_{, . . . , n}′ _vertically

below. See Fig. 1 for the graph of a block bijection θ ∈ I∗

8 with domain

(1,2|3|4,6,7|5,8) and range (1|2,4|3|5,6,7,8). In an obvious notation, we also writeθ= 1₂,_,2₄

₅_,₆3_,₇_,₈ 4,6₁,7

5₃,8

[image:2.595.244.367.400.460.2]

.

Figure 1: A graphical representation of a block bijectionθ∈I∗

8.

To multiply two such diagrams, they are stacked vertically, with the “interior” rows of vertices coinciding; then the connected components of the resulting graph are constructed and the interior vertices are ignored. See Fig. 2 for an example.

It is clear from its graphical representation that I∗

n is a submonoid of the

partition monoid, though not one of the submonoids discussed in [3]. Maltcev [5] shows thatI∗

n with the zero of the partition monoid adjoined is a maximal

inverse subsemigroup of the partition monoid, and gives a set of generators for

I∗

n. These generators are completely regular; later in this paper, we present

auxiliary results on the generation of inverse semigroups by completely regular elements. Although these results are of interest in their own right, our main goal is to obtain a presentation, in terms of generators and relations, of I∗

n.

Our method makes use of known presentations of some special subsemigroups of I_n∗_{. We now describe these subsemigroups, postponing their presentations}

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θ1=

θ2=

[image:3.595.143.473.135.226.2]

=θ1θ2

Figure 2: The product of two block bijectionsθ1, θ2∈I4∗.

The group of units of I∗

n is the symmetric group Sn, while the semilattice

of idempotents is (isomorphic to) E_n_{, the set of all equivalences on} n, with multiplication being join of equivalences. Another subsemigroup consists of those block bijections which are induced by permutations of n acting on the equivalence relations; this is thefactorizable part of I_n∗_{, which we denote by} F_n_{, and which is equal to the set product} E_n_S_n ₌_S_nE_n_{. In [2] these elements}

were calleduniform, and in [5]type-preserving, since they have the characteristic property that corresponding blocks are of equal cardinality. We will also refer to the local submonoid εI∗

Xε ofIX∗ determined by a non-identity idempotent

ε. This subsemigroup consists of all β ∈I∗

X for whichε is a (left and right)

identity. Recalling that the idempotent ε is an equivalence on X, it is easy to see that there is a natural isomorphism εI∗

Xε → IX/ε∗ . As an example

which we make use of later, whenX =n and ε= (1,2|3| · · · |n), we obtain an isomorphism Υ : εI∗

nε → In∗−1. Diagrammatically, we obtain a graph of

βΥ∈I∗

n−1from a graph ofβ ∈εIn∗εby identifying vertices 1≡2 and 1′ ≡2′,

relabelling the vertices, and adjusting the edges accordingly; an example is given in Fig. 3.

β = =βΥ

Figure 3: The action of the map Υ :εI∗

[image:3.595.174.428.501.555.2]

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1.2 Presentations

LetX be an alphabet (a set whose elements are calledletters), and denote by X∗_{the free monoid on}_{X. For}_R_⊆_X∗_×_X∗_{we denote by}_R♯_{the congruence on}

X∗ _{generated by} _{R, and we define}_h_X_|_R_i₌_X∗_/R♯_{. We say that a monoid}_M

has presentationhX|RiifM ∼=hX|Ri. Elements ofX andRare called genera-tors and relations (respectively), and a relation (w1, w2)∈Ris conventionally

displayed as an equation: w1=w2. We will often make use of the following

universal property of hX|Ri. We say that a monoid S satisfies R (or that R

holds in S) via a mapiS :X →S if for all (w1, w2)∈Rwe havew1i∗S =w2i∗S

(wherei∗

S :X∗→S is the natural extension of iS toX∗). ThenM =hX |Ri

is the monoid, unique up to isomorphism, which is universal with respect to the property that it satisfiesR (via iM :x7→xR♯); that is, if a monoidS satisfies

RviaiS, there is aunique homomorphismφ:M →S such thatiMφ=iS:

S

X iM M =hX|Ri

iS

φ

This mapφ is called thecanonical homomorphism. If X generates S viaiS,

thenφis surjective sincei∗

S is.

2 Inverse Monoids Generated by Completely

Reg-ular Elements

In this section we present two general results which give necessary and sufficient conditions for a monoid generated by completely regular elements to be inverse, with a semilattice of idempotents specified by the generators.

For a monoidS, we writeE(S) andG(S) for the set of idempotents and group of units ofS (respectively). Suppose now thatS is an inverse monoid (so that E(S) is in fact a semilattice). Thefactorizable part ofSisF(S) =E(S)G(S) = G(S)E(S), and S is factorizable if S = F(S); in general, F(S) is the largest factorizable inverse submonoid ofS.

Recall that an element x of a monoid S is said to be completely regular if its H_-class _H_x _{is a group.} _{For a completely regular element} _x _∈ _{S, we}

write x−1 _{for the inverse of} _x _in _H

x, and x0 for the identity element of Hx.

Thus, xx−1₌_x−1_x₌_x0 _{and, of course,} _x0 _∈ _{E(S). If} _X _⊆ _{S, we write}

X0₌

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Proposition 1 LetS be a monoid, and suppose thatS=hXiwith eachx∈X

completely regular. ThenS is inverse with E(S) = hX0_i_{if and only if, for all}

x, y∈X,

(i) x0_y0₌_y0_x0_{, and}

(ii) y−1_x0_y_{∈ h}_X0_i_.

Proof IfS is inverse, then (i) holds. Also, forx, y∈X, we have

(y−1_x0_y)2₌_y−1_x0_y0_x0_y₌_y−1_x0_x0_y0_y₌_y−1_x0_y,

so thaty−1_x0_y_∈_{E(S). So if}_{E(S) =}_h_X0_i_{, then (ii) holds.}

Conversely, suppose that (i) and (ii) hold. From (i) we see thathX0_{i ⊆}_E(S)

and that hX0_i _{is a semilattice. Now let} _y _∈ _X_{. Next we demonstrate, by}

induction onn, that

y−1(X0)ny⊆ hX0i, (A)

for alln∈N_{. Clearly (A) holds for}_n_{= 0. Suppose next that (A) holds for some}

n∈N_{, and that}_w_∈_(X0₎n+1_{. So}_w₌_x0_v_{for some}_x_∈_X _and_v_∈_(X0₎n_{. But}

then by (i) we have

y−1_wy₌_y−1_x0_vy₌_y−1_y0_x0_vy₌_y−1_x0_y0_vy_{= (y}−1_x0_y)(y−1_vy).

Condition (ii) and an inductive hypothesis that y−1_vy _{∈ h}_X0_i _{then imply}

y−1_wy_{∈ h}_X0_i_{. So (A) holds.}

Next we claim that for eachw∈S there exists w′_∈_S _{such that}

w′_h_X0_i_w_{⊆ h}_X0_i_, _(B1)

ww′, w′w∈ hX0i, (B2)

ww′w=w, w′ww′ =w′. (B3)

We prove the claim by induction on thelengthofw(that is, the minimal value ofn ∈N _{for which} _w_∈ _Xn_{). The case} _n_{= 0 is trivial since then}_w _{= 1 and}

we may takew′ _{= 1. Next suppose that} _n_∈_N _{and that the claim is true for}

elements of lengthn. Suppose thatw∈S has length n+ 1, so thatw=xvfor somex∈X andv∈S of lengthn. Putw′ ₌_v′_x−1_{. Then}

w′hX0iw=v′x−1hX0ixv⊆v′hX0iv⊆ hX0i,

the first inclusion holding by (A) above, and the second by inductive hypothesis. Thus (B1) holds. Also,

ww′ ₌_xvv′_x−1_∈_x_h_X0_i_x−1_{⊆ h}_X0_i

and

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by (A), (B1), and the induction hypothesis, establishing (B2). For (B3), we have

ww′w=xvv′x−1xv=xvv′x0v=xx0vv′v=xv=w, using (B2), (i), and the inductive hypothesis. Similarly we have

w′ww′=v′x0vv′x−1=v′vv′x0x−1=v′x−1=w′, completing the proof of (B3).

Since S is regular, by (B3), the proof will be complete if we can show that E(S)⊆ hX0_i_{. So suppose that} _w_∈_{E(S), and choose} _w′ _∈_S _{for which (B1—}

B3) hold. Then

w′=w′ww′= (w′w)(ww′)∈ hX0i

by (B2), whencew′_∈_{E(S). But then}

w=ww′w= (ww′)(w′w)∈ hX0i,

again by (B2). This completes the proof. 2

Proposition 2 Suppose that S is a monoid and that S = hG∪ {z}i where

G=G(S)andz3₌_z_{. Then}_S _{is inverse with}

E(S) =hg−1z2g | g∈Gi and F(S) =hG∪ {z2}i

if and only if, for allg∈G,

g−1z2gz2=z2g−1z2g (C1) zg−1z2gz∈ hG∪ {z2}i. (C2)

Proof First observe that z is completely regular, with z =z−1 _and_z0 ₌_z2_.

Now put

X=G∪ {g−1_zg_|_g_∈_G_}_.

Then S =hXi, and each x∈X is completely regular. Further, if y =g−1_zg

(withg∈G), theny−1₌_y_and_y0₌_y2₌_g−1_z2_{g. Thus,}_X0₌_{₁_{} ∪ {}_g−1_z2_g _|_g_∈_G_}_.

Now if S is inverse, then (C1) holds. Also, zg−1_z2_gz _∈_{E(S) for all}_g _∈_G _so

that (C2) holds ifF(S) =hG∪ {z2_}i_.

Conversely, suppose now that (C1) and (C2) hold. We wish to verify Conditions (i) and (ii) of Proposition 1, so let x, y∈ X. If x0 _{= 1 or}_y0 _{= 1, then (i) is}

immediate, so suppose x0₌_g−1_z2_g _and _y0 ₌ _h−1_z2_h _(where _{g, h} _∈ _{G). By}

(C1) we have

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and (i) holds.

If x0 _{= 1 or} _y _∈ _{G, then (ii) is immediate, so suppose} _x0 ₌ _g−1_z2_g _and

y=h−1_zh_(where_{g, h}_∈_{G). Then}_y₌_y−1 _{and, by (C2),}

y−1x0y=h−1(zhg−1z2gh−1z)h∈h−1hG∪ {z2}ih⊆ hG∪ {z2}i. But by [1, Lemma 2] and (C1),hG∪ {z2_}i_{is a factorizable inverse submonoid}

ofS withE hG∪ {z2_}i

=hX0_i_{. Since}

(y−1_x0_y)2₌_y−1_x0_y0_x0_y₌_y−1_x0_y_∈_E _h_G_{∪ {}_z2_}i ,

it follows that y−1_x0_y _{∈ h}_X0_i_{, so that (ii) holds. So, by Proposition 1,} _S _is

inverse withE(S) = X0

=

g−1z2g |g∈G

and, moreover, its factorizable part satisfies

F(S) =E(S)G⊆ hG∪X0i ⊆ hG∪ {z2}i ⊆F(S). HenceF(S) =hG∪ {z2_}i_{, and the proof is complete.}

2

3 A Presentation of

I

_n∗

Ifn≤2, thenI∗

n =Fn is equal to its factorizable part. A presentation ofFn

(for anyn) may be found in [1] so, without loss of generality, we will assume for the remainder of the article thatn≥3. We first fix an alphabet

X ₌X_n₌_{_{x, s}₁_{, . . . , s}_n₋₁_}_.

Several notational conventions will prove helpful, and we note them here. The empty word will be denoted by 1. A wordsi· · ·sj is assumed to be empty if

either (i)i > jand the subscripts are understood to be ascending, or (ii) ifi < j and the subscripts are understood to be descending.

For 1≤i, j≤n−1, we define integers

mij =

  

 

1 if i=j 3 if |i−j|= 1 2 if |i−j|>1.

It will be convenient to use abbreviations for certain words in the generators which will occur frequently in relations and proofs. Namely, we write

σ=s2s3s1s2,

and inductively we define wordsl2, . . . , ln−1 andy3, . . . , yn by

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and

y3=x and yi+1=liyisi for 3≤i≤n−1.

Consider now the setR₌R_n _{of relations}

(sisj)mij = 1 for 1≤i≤j ≤n−1 (R1)

x3=x (R2)

xs1=s1x=x (R3)

xs2x=xs2xs2=s2xs2x

=xs2x2=x2s2x (R4)

x2σx2σ=σx2σx2=xs2s3s2x (R5)

yisiyi=siyisi for 3≤i≤n−1 (R6)

xsi=six for 4≤i≤n−1. (R7)

Before we proceed, some words of clarification are in order. We say a relation belongs to R_n _vacuously _{if it involves a generator} _s_i _{which does not belong}

to X_n_{; for example, (R5) is vacuously present if} _n _{= 3 because} X₃ _{does not}

contain the generators3. So the reader might like to think ofRn as the set of

relations (R1—R4) ifn = 3, (R1—R6) if n= 4, and (R1—R7) if n≥ 5. We also note that we will mostly refer only to thei= 3 case of relation (R6), which simply saysxs3x=s3xs3.

We aim to show that I∗

n has presentation hX | Ri, so put M = Mn =

hX _|R_i₌X∗_/_R♯_{. Elements of}_M _are_R♯_{-classes of words over}_X_{. However,}

in order to avoid cumbersome notation, we will think of elements ofM simply as words overX_{, identifying two words if they are equivalent under the relations} R_{. Thus, the reader should be aware of this when reading statements such as}

“Let w∈M” and so on.

With our goal in mind, consider the map

Φ = Φn :X →In∗

defined by

xΦ = 1₃,2 ₁3_,₂

4₄ ···_··· n_n

and siΦ = 1₁

···_···

i_i−₋1₁

_i₊₁i

i+1_i

i_i+2₊₂

···_···

n_n

for 1≤i≤n−1.

3.1 The factorizable part of

I

∗

n

Define an alphabetX_F ₌_{_{t, s}₁_{, . . . , s}_n₋₁_}_{, and consider the set}R_F _{of relations}

(sisj)mij = 1 for 1≤i≤j≤n−1 (F1)

t2₌_t _(F2)

ts1=s1t=t (F3)

tsi =sit for 3≤i≤n−1 (F4)

ts2ts2=s2ts2t (F5)

tσtσ =σtσt. (F6)

(Recall thatσdenotes the word s2s3s1s2.) The following result was proved in

[1].

Theorem 2 The monoidF_n₌_F₍I∗

n) has presentationhXF |RFivia

si 7→siΦ, t7→(1,2|3| · · · |n).2

Lemma 3 The relationsR_F _{hold in} _M _{via the map}_{Θ :}_t_7→_x2_{, s}

i7→si.

Proof Relations (F1—F3) are immediate from (R1—R3); (F5) follows from several applications of (R4); and (F6) forms part of (R5). The i ≥ 4 case of (F4) follows from (R7), and thei= 3 case follows from (R1) and (R6), since

x2_s

3=xs3s3xs3=xs3xs3x=s3xs3s3x=s3x2.

2

It follows that Θ◦Φ extends to a homomorphism of hX_F _|R_F_i_toF_n_{, which}

is an isomorphism by Theorem 2. We conclude that Φ|hx2_,s

1,...,sn−1i= Φ|im(Θ) is injective (and therefore an isomorphism).

3.2 The structure of

_M

It is easy to see that the group of units G(M) is the subgroup generated by

{s1, . . . , sn−1}. The reason for this is that relations (R2—R7) contain at least

one occurrence ofxon both sides. Now (R1) forms the set of defining relations in Moore’s famous presentation [6] of the symmetric groupSn. Thus we may

identifyG(M) withSn in the obvious way. Part (i) of the following well-known

result (Lemma 8) gives a normal form for the elements ofSn(and is probably due

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from the first, and is expressed in terms of a convenient contracted notation which is defined as follows. Let 1≤i≤n−1, and 0≤k≤n−1. We write

ski =

(

si if i≤k

1 otherwise.

The reader might like to think of this as abbreviating sk_i≥i, where k ≥i is a boolean value, equal to 1 ifk≥iholds and 0 otherwise.

Lemma 4 Letg∈G(M) =hs1, . . . , sn−1i. Then

(i) g= (si1· · ·sj1)· · ·(sik· · ·sjk)for somek≥0and somei1≤j1, . . . , ik ≤jk

with1≤ik <· · ·< i1≤n−1, and

(ii) g =hsk

2sk3sk4(s5· · ·sk)s1ℓsℓ2sℓ3(s4· · ·sℓ) =hsk2sk3sk4sℓ1sℓ2sℓ3(s5· · ·sk)(s4· · ·sℓ)

for someh∈ hs3, . . . , sn−1i,k≥1 andℓ≥0. 2

We are now ready to prove the main result of this section.

Proposition 5 The monoidM =hX _|R_i_{is inverse, and we have}

E(M) =hg−1x2g |g∈G(M)i and F(M) =hx2, s1, . . . , sn−1i.

Proof PutG=G(M) =hs1, . . . , sn−1i. SoM =hG∪{x}iandx=x3. We will

now verify conditions (C1) and (C2) of Proposition 2. By Lemma 3,hG∪ {x2_}i

is a homomorphic (in fact isomorphic) image ofF_n_{, so}_g−1_x2_g _{commutes with}

x2 _{for all}_g_∈_{G, and condition (C1) is verified.}

To prove (C2), letg∈G. By Lemma 4, we have

g=hsk2sk3sk4sℓ1sℓ2sℓ3(s5· · ·sk)(s4· · ·sℓ)

for someh∈ hs3, . . . , sn−1i,k≥1 andℓ≥0. Now

xg−1x2gx

=x(sℓ· · ·s4)(sk· · ·s5)sℓ3sℓ2sℓ1sk4sk3sk2(h−1x2h)sk2sk3sk4sℓ1sℓ2sℓ3(s5· · ·sk)(s4· · ·sℓ)x

= (sℓ· · ·s4)(sk· · ·s5)xsℓ3sℓ2sℓ1sk4sk3sk2x2sk2sk3sk4sℓ1s2ℓsℓ3x(s5· · ·sk)(s4· · ·sℓ),

by (R7), (F4), and (R1). Thus it suffices to show that x(x2₎π_x_{∈ h}_G_{∪ {}_x2_}i_,

where we have writtenπ=sk

2sk3sk4sℓ1sℓ2sℓ3. Altogether there are 16 cases to

con-sider for all pairs (k, ℓ) withk= 1,2,3,≥4 andℓ= 0,1,2,≥3. Table 1 below contains an equivalent form of x(x2₎π_x _{as a word over} _{_x2_{, s}

1, . . . , sn−1} for

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ℓ= 0 ℓ= 1 ℓ= 2 ℓ≥3

x2

s2x2 x2σx2σ

k= 1 (R2) (R2,3) (R3,4) (R1,2,3,5) and (F4)

x2_s

2x2 x2s2x2 x2s2x2 x2σx2σ

k= 2 (R4) (R3,4) (R1,3,4) (R1,3) and (k, ℓ) = (1,≥3)

x2

σx2

σ x2

σx2

σ x2

s2s3s2x2 x2s2s3s2x2

k= 3 (k, ℓ) = (1,≥3) (R3) and (R1,2,5) (R1,3) and (k, ℓ) = (1,≥3) (k, ℓ) = (3,2)

s4x2σx2σs4 s4x2σx2σs4 s4x2s2s3s2x2s4 s3s4x2σx2σs4s3

[image:13.595.144.463.122.253.2]

k≥4 (R7) and (R3) and (R1,7) and (R1,2,5,6,7) (k, ℓ) = (1,≥3) (k, ℓ) = (≥4,0) (k, ℓ) = (3,2) and (F4)

Table 1: Expressions for x(x2₎π_x _{and the relations used. See text for further}

explanation.

from left to right, then the second, third, and fourth rows. Thus, as for example in the case (k, ℓ) = (2,1), we have used expressions from previously considered cases.

In order that readers need not perform all the calculations themselves, we pro-vide a small number of sample derivations. The first case we consider is that in which (k, ℓ) = (1,2). In this case we have π=s1s2 and, by (R3) and several

applications of (R4), we calculate

x(x2)πx=xs2s1x2s1s2x=xs2x2s2x=x2s2x2.

Next suppose (k, ℓ) = (1,≥ 3). (Here we mean that k= 1 and ℓ≥ 3.) Then π=s1s2s3 and

x(x2₎π_x₌_xs

3s2s1x2s1s2s3x

=xs3s2x2s2s3x by (R3)

=xs3s2s3x2s3s2s3x by (R1) and (F4)

= (x2σx2σ)(σx2σx2) by (R1) and (R5)

=x2σx2σx2 by (R1) and (R2)

=x2σx2σ by (R5) and (R2).

If (k, ℓ) = (3,2), thenπ=π−1₌_σ_{by (R1) and, by (R2) and (R5), we have}

x(x2)πx=xσx2σx=x(σx2σx2)x=x(xs2s3s2x)x.

If (k, ℓ) = (≥4,2) thenπ=s2s3s4s1s2=σs4 by (R1), and so, using (R7) and

the (k, ℓ) = (3,2) case, we have

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Finally, we consider the (k, ℓ) = (≥4,≥3) case. Here we haveπ=s2s3s4s1s2s3=

σs4s3 by (R1), and so

x(x2)πx=xs3s4σx2σs4s3x

=xx2_s

3s4σx2σs4s3x by (R2)

=xs3s4x2σx2σs4s3x by (F4)

=xs3s4xs2s3s2xs4s3x by (R5)

=xs3xs4s2s3s2s4xs3x by (R7)

=s3xs3s4s3s2s3s4s3xs3 by (R1) and (R6)

=s3xs4s3s4s2s4s3s4xs3 by (R1)

=s3s4xs3s2s3xs4s3 by (R1) and (R7)

=s3s4(x2σx2σ)s4s3 by (R1) and (R5).

After checking the other cases, the proof is complete. 2

After the proof of Lemma 3, we observed that Φ|hx2_,s

1,...,sn−1i is injective. By Proposition 5, we conclude that Φ|F(M)is injective. In particular, both Φ|E(M)

and Φ|G(M) are injective.

3.3 A local submonoid

Now pute = x2 _∈ _M_{. So clearly} _e _{is a non-identity idempotent of} _M_{. Our}

goal in this section is to show that e has property (P), and that eM e is a homomorphic image ofMn−1.

Lemma 5 The non-identity idempotente=x2_∈_M _{has property (P).}

Proof Let 1 6= f ∈ E(M). By Proposition 5 we have f = eg1eg2· · ·egk for

somek≥1 andg1, g2, . . . , gk ∈G(M). But then

fg−1

1 =e eg2g−11· · ·egkg1−1e∈eM e.

2

We now define words

X =s3xσxs3, S1=x, and Sj =esj+1 for j= 2, . . . , n−2,

and write Y ₌Y_n₋₁ ₌_{_{X, S}₁_{, . . . , S}_n₋₂_}_{. We note that}_e _{is a left and right}

identity for the elements ofY _{so that}Y _⊆_{eM e, and that}_X ₌_y₄ _(by

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Proposition 6 The submonoid eM e is generated (as a monoid with identity

e) byY_.

Proof We take w ∈ M with the intention of showing that u = ewe ∈ eM e belongs tohY_i_{. We do this by induction on the (minimum) number}_d₌_d(w)

of occurrences ofxδ _(δ_{∈ {}_1,₂_}_{) in}_{w. Suppose first that}_d_{= 0, so that}_u₌_ege

whereg∈G(M). By Lemma 4 we have

g=h(sj2s

j

3si1si2)(s4· · ·sj)(s3· · ·si)

for some h∈ hs3, . . . , sn−1iand j ≥1, i ≥0. Put h′ = (s4· · ·sj)(s3· · ·si)∈

hs3, . . . , sn−1i. Now by (F2) and (F4) we have

u=ege=eh·e(sj2s

j

3si1si2)e·eh′.

By (F2) and (F4) again, we see thateh, eh′ ∈ hS2, . . . , Sn−1i, so it is sufficient

to show that the wordeπebelongs tohY_i_{, where we have written}_π₌_sj

2s

j

[image:15.595.195.419.336.437.2]

3si1si2.

Table 2 below contains an equivalent form of eπe as a word over Y _{for each}

(i, j), as well as a list of the relations used in deriving the expression.

j= 1 j= 2 j≥3

e X2

X2

S2

i= 0 (R2) (R1,2,5) (R2), (F4), and and (F4) (i, j) = (0,2)

e X2

X2

S2

i= 1 (R2,3) (R3) and (R3) and (i, j) = (0,2) (i, j) = (0,≥3)

X2

S1S2XS2S1

i≥2 (R3) and (R1,3) and (R1,2) (i, j) = (0,2) (i, j) = (0,2) and (F4)

Table 2: Expressions foreπeand the relations used. See text for further expla-nation.

Most of these derivations are rather straightforward, but we include two example calculations. For the (i, j) = (0,2) case, note that

X2=s3xσxs3s3xσxs3=s3x(x2σx2σ)xs3 by (R1) and (R2)

=s3x(xs2s3s2x)xs3 by (R5)

=s3x2s3s2s3x2s3 by (R1)

=x2s2x2 by (F4) and (R1)

=eπe.

For the (i, j) = (≥2,≥3) case, we have π=σand

eπe=x2σx2=xs3s3xσxs3s3x by (R1)

=x(x2s3)s3xσxs3(s3x2)x by (R2)

=x(x2s3)s3xσxs3(x2s3)x by (F4)

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This establishes thed= 0 case. Now supposed≥1, so thatw=vxδ_g_{for some}

δ∈ {1,2},v∈M withd(v) =d(w)−1, andg∈G(M). Then by (R2),

u=ewe= (eve)xδ(ege).

Now xδ _{belongs to}_h_Y_i _since_xδ _{is equal to} _S

1 (ifδ = 1) or e(if δ = 2). By

an induction hypothesis we haveeve∈ hY_i_{and, by the}_d_{= 0 case considered}

above, we also haveege∈ hY_i_. ₂

The next step in our argument is to prove (in Proposition 7 below) that the elements of Y_n₋₁ _{satisfy the relations} R_n₋₁ _{via the obviously defined map.}

Before we do this, however, it will be convenient to prove the following basic lemma. Ifw∈M, we write rev(w) for the word obtained by writing the letters ofwin reverse order. We say thatwissymmetric ifw= rev(w).

Lemma 6 Ifw∈M is symmetric, then w=w3 _and_w2_∈_E(M₎_.

Proof Now z =z−1 _{for all}_x_∈ _X _{and it follows that} _w−1 _{= rev(w) for all}

w∈M. So, ifwis symmetric, thenw=w−1_{, in which case}_w₌_ww−1_w₌_w3_.

2

Proposition 7 The elements ofY_n₋₁ _{satisfy the relations}R_n₋₁ _{via the map}

Ψ :X_n₋₁_→_{eM e}_:_x_7→_{X, s}_i_7→_S_i_.

Proof We consider the relations fromR_n₋₁ _{one at a time. In order to avoid}

confusion, we will refer to the relations fromR_n₋₁_{as (R1)}′_{, (R2)}′_{, etc. We also}

extend the use of upper case symbols for the element Σ =S2S3S1S2 as well as

the wordsLi (for i = 2, . . . , n−2) and Yj (for j = 3, . . . , n−1). It will also

prove convenient to refer to the idempotents

ei=e(s2···si)(s1···si−1)∈E(M),

defined for each 1 ≤ i ≤ n. Note that e1 = e, and that eiΦ ∈ In∗ is the

idempotent with domain (1| · · · |i−1|i, i+ 1|i+ 2| · · · |n).

We first consider relation (R1)′_{. We must show that (S}

iSj)mij = e for all

1≤i≤j ≤n−2. Suppose first thati =j. NowS2

1 =e by definition and if

2≤i≤n−2 then, by (R1), (R2), (R7), and (F4), we haveS2

i =esi+1esi+1 =

e2s2i+1=e. Next, if 2≤j≤n−2, then

(S1Sj)m1j = (xesj+1)m1j = (xsj+1)m1j

= (

xs3xs3xs3=s3xs3s3xs3=s3x2s3=x2=e if j= 2

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by (R1), (R2), (R6), (R7), and (F4). Finally, if 2≤i≤j≤n−2, then

(SiSj)mij = (esi+1esj+1)mi+1,j+1 =e(si+1sj+1)mi+1,j+1 =e,

by (R1), (R2), and (F4). This completes the proof for (R1)′_.

For (R2)′_{, we have}

X3_{= (s}

3xσxs3)(s3xσxs3)(s3xσxs3)

=s3xσx2σx2σxs3 by (R1)

=s3xx2σx2σσxs3 by (R5)

=s3xσxs3 by (R1) and (R2)

=X.

For (R3)′, first note that

XS1=s3xσxs3x

=s3xσs3xs3 by (R6)

=s3xs1σxs3 by (R1)

=s3xσxs3 by (R3)

=X.

(Here, and later, we use the fact thatσs3=s1σ, and so alsoσs1=s3σ. These

are easily checked using (R1) or by drawing pictures.) The relationS1X =X

is proved by a symmetrical argument.

To prove (R4)′_{, we need to show that}_XS

2X is a (left and right) zero for X

and S2. Since X, S2 ∈ hx, s1, s2, s3i, it suffices to show that XS2X is a zero

for each of x, s1, s2, s3. In order to contract the proof, it will be convenient to

use the following “arrow notation”. Ifaandbare elements of a semigroup, we writea >−banda−−> bto denote the relationsab=a(ais a left zero forb) and ba=a(ais a right zero forb), respectively. The arrows may be superimposed, so thata >−−> bindicates the presence of both relations. We first calculate

XS2X = (s3xσxs3)es3(s3xσxs3)

=s3xσxs3xσxs3 by (R1) and (R2)

=s3xσs3xs3σxs3 by (R6)

=s3xs1σxσs1xs3 by (R1)

=s3xσxσxs3 by (R3).

Putw=s3xσxσxs3. We see immediately thatw >−xsince

wx=s3xσxσxs3x=s3xσxσs3xs3=s3xσxs1σxs3=s3xσxσxs3=w,

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6. Since>−−>is transitive, the proof of (R4)′ _{will be complete if we can show}

thatw2_>₋₋_{> s}

1, s2, s3. Now by Lemma 6 again we havew2∈E(M) and, since

w2_{Φ = (e}

1e2e3)Φ as may easily be checked diagrammatically, we have w2 =

e1e2e3 by the injectivity of Φ|E(M). But (e1e2e3)Φ>−−> s1Φ, s2Φ, s3Φ in Fn

and so, by the injectivity of Φ|F(M), it follows thatw2=e1e2e3>−−> s1, s2, s3.

Relations (R5—R7)′ _{all hold vacuously if} _n _{= 4, so for the remainder of the}

proof we assume thatn≥5.

Next we consider (R5)′_{. Now, by (R2) and (F4), we see that}

Σ =S2S3S1S2= (es3)(es4)x(es3) =s3s4xs3.

In particular, Σ is symmetric, by (R7), and Σ−1 _{= Σ. Also,} _X ₌ _s

3xσxs3 is

symmetric and soX2_{is idempotent by Lemma 6. It follows that}_X2_ΣX2_{Σ and}

ΣX2_ΣX2 _{are both idempotent. Since (X}2_ΣX2_{Σ)Φ = (ΣX}2_ΣX2_{)Φ, as may}

easily be checked diagrammatically, we conclude that X2ΣX2Σ = ΣX2ΣX2, by the injectivity of Φ|E(M). It remains to check thatXS2S3S2X = ΣX2ΣX2.

Since (XS2S3S2X)Φ = (ΣX2ΣX2)Φ, it suffices to show that XS2S3S2X ∈

E(M). By (R1), (R2), and (F4),

XS2S3S2X= (s3xσxs3)es3es4es3(s3xσxs3) =s3(xσx)s4(xσx)s3,

so it is enough to show thatv= (xσx)s4(xσx) is idempotent. We see that

v2=xσxs4xσx2σxs4xσx

=xσxs4x(x2σx2σ)xs4xσx by (R2)

=xσxs4x(xs2s3s2x)xs4xσx by (R5)

=xσxs4s2s3s2s4xσx by (R2) and (R7).

Put u = xσx. Since u is symmetric, Lemma 6 says that u2 _∈ _E(M_{). One}

verifies easily that (u2_s

4s2s3s2s4u2)Φ = (u2s4u2)Φ inFn and it follows, by the

injectivity of Φ|F(M), that u2s4s2s3s2s4u2 =u2s4u2. By Lemma 6 again, we

also haveu=u3 _{so that}

v2=us4s2s3s2s4u=uu2s4s2s3s2s4u2u=uu2s4u2u=us4u=v,

and (R5)′ _holds.

Now we consider (R6)′_{, which says} _Y

iSiYi = SiYiSi for i ≥ 3. So we must

calculate the wordsYiwhich, in turn, are defined in terms of the wordsLi. Now

L2=XS2S1 and Li+1=Si+1LiSi+1Si fori≥2. A straightforward induction

shows that Li=li+1e for all i≥2. This, together with the definition of the

words Yi (as Y3 = X, and Yi+1=LiYiSi for i ≥ 3) and a simple induction,

shows thatYi=yi+1 for alli≥3. But then for 3≤i≤n−2, we have

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Here we have used (R6) and (F4), and the fact, verifiable by a simple induction, thatyje=eyj=yj for allj.

Relation (R7)′ _{holds vacuously when}_n_{= 5 so, to complete the proof, suppose}

n ≥ 6 and i ≥ 4. Now Xe = eX = X as we have already observed, and Xsi+1 =si+1X by (R1) and (R7). Thus X commutes with esi+1 =Si. This

completes the proof. 2

3.4 Conclusion

We are now ready to tie together all the loose ends.

Theorem 7 The dual symmetric inverse monoidI∗

n has presentationhX |Ri

viaΦ.

Proof All that remains is to show that Φ = Φn is injective. In Proposition

3 we saw that this was true forn = 3, so suppose that n≥ 4 and that Φn−1

is injective. By checking that both maps agree on the elements ofX_n₋₁_{, it is}

easy to see that Ψ◦Φ|eM e◦Υ = Φn−1. (The map Υ was defined at the end of

Section 1.1., and Ψ in Proposition 13.) Now Ψ is surjective (by Proposition 6) and Φn−1is injective (by assumption), so it follows that Φ|eM eis injective. After

the proof of Proposition 5, we observed that Φ|E(M) and Φ|G(M)are injective.

By Lemma 5, e has property (P) and it follows, by Proposition 4, that Φ is

injective. 2

We remark that the method of Propositions 5 and 13 may be used to provide a concise proof of the presentation ofI_n _{originally found by Popova [7].}

References

[1] D. G. FitzGerald, A presentation for the monoid of uniform block permuta-tions,Bull. Austral. Math. Soc.68(2003) 317–324.

[2] D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory,J. Austral. Math. Soc. (Series A)64(1998) 345–367.

[3] T. Halverson and A. Ram, Partition algebras, European J. Combin. 26

(2005) 869–921.

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[5] Victor Maltcev, On a new approach to the dual symmetric inverse monoid

I∗

X, Int. J. Algebr. Comput.17 (2007) 567–591.

[6] E. H. Moore, Concerning the abstract groups of order k! and (1/2)k! holohedrically isomorphic with the symmetric and alternating substitution groups on k letters,Proc. London Math. Soc.28(1897) 357–366.