Catalan monoids, monoids of local endomorphisms, and ... - CiteSeerX

Catalan monoids, monoids of local endomorphisms, and their presentations Andrew Solomon

email: [email protected]

Abstract

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of nite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Azenstat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a nite symmetric chain.

1 Introduction This paper is entirely concerned with the Catalan monoid of a directed graph, introduced by the author in [16]. We take this opportunity, in Section 3, to give some justi cation for its naming, and to de ne the partial Catalan monoid of a directed graph. Certain Catalan monoids have already been studied extensively, in particular, those which consist of order-preserving transformations of [n], where [n] denotes the set f1; : : : ; ng. Monoids of order-preserving transformations of [n] are studied by Azenstat [1] and Popova [14] who nd presentations; Gomes and Howie [4], Howie [10] and Higgins [5] who give combinatorial results; and Higgins [6] and Vernitskii and Volkov [18] who discuss their divisors.

2 Preliminaries

Let  be a (partial) transformation of a set X . Denote by
()  X the domain of . If
() = X we shall say that  is a full transformation of X . The shift of  is the set Sh() = fx 2 X j x 2=
() or x 6= xg.

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A directed graph G is a pair (V (G); E (G)) where E (G)  V (G)  V (G). We write x ! y if (x; y) 2 E (G). Call V (G) the set of vertices of G and E (G) the set of edges of G. Say that G is a symmetric graph if ! is a symmetric relation on V (G). Unless otherwise stated, the word `graph' shall mean a directed graph, and in the sequel we will consider only graphs G which have the following properties : V (G) is countable; each vertex of G has nite in{degree and out{degree; and G has no loops, where a loop is an edge x ! x for some x 2 V (G). A path from x to y for x; y 2 V (G) is a nite sequence of edges x ! x1 !


! xn = y where n  0. We say that the path thus described has length n. If n = 0 we say that the path is trivial. This should be distinguished from a loop on x 2 V (G) which is a path of length 1 from x to x. A cycle on x 2 G is a path from x to x of length n > 1 where xi 6= x for any 1  i < n. Say that a graph is acyclic if it has no cycles. It is clear that an acyclic graph gives rise to a partial order on the set of vertices by de ning x  y if there is a path from x to y. Let G be a graph and  a transformation of V (G). We say that  is extensive if, for each x 2
(), there is a (possibly trivial) path from x to x. In an acyclic graph, extensive transformations are also called non-increasing since x  x for all x 2
(). If G is acyclic and for each x; y 2
() such that x  y we also have x  y, then we say that  is order-preserving or an order-endomorphism.

3 Catalan Monoids

De nition 3.1 For each edge s = (a; b) 2 E (G), let fs label the transformation fs of V (G) de ned by a 7?! b x 7?! x if x 6= a, called the elementary transition of s. Let the set of all such labels be denoted by G . We de ne the Catalan monoid C (G) of G to be the monoid of transformations of V (G) generated by the elementary transitions. Remark 3.2 In [12] Molchanov cites the work of Filippov [3] who studies the partial groupoid Ex(G) of transformations  of V (G) such that for each x 2 V (G) either x = x or (x; x) 2 E (G). Note that the Catalan monoid of G is not simply the monoid closure of Ex(G). For example, Ex(1 $ 2) contains the permutation which interchanges vertices 1 and 2, where C (1 $ 2) does not. Example 3.3 It is clear that if G is the graph 1 2


n then every element of C (G) is an order-preserving, non-increasing transformation of [n]. In fact we shall see in the next section (Corollary 4.13) that C (G) is precisely the monoid of orderpreserving, non-increasing transformations of [n]. Under this assumption, we are able

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to oer an alternative proof of the following proposition, which appears in [5], and justi es the name `Catalan monoid'.

Proposition 3.4 Let G be the graph 1 2


n. Then the cardinality of C (G) is the nth Catalan number (de ned in [15], page 101). Proof. If we write an element  of C (G) as the pair of sequences M1 < M2 <


< Mk and q1 < q2 <


< qk where Mi is the top element of the kernel class of points which map to qi under  (an example of which is illustrated below) M1

q1=1

M2

M3

q2

Mk-1

q3

qk-1

Mk=n

qk

then it is easy to see that C (G) is in bijection with the set of balanced n-bracketings by  7! (M1 )q2?q1 (M2?M1 )q3?q2 : : :)qk ?qk?1 (Mk ?Mk?1 )n+1?qk where (q and )q denote (|:{z: : }( and )| :{z: :)} respectively. Thus the number of elements of q

q

C (G) is the nth Catalan number. De nition 3.5 For each vertex x 2 V (G), let ex label the (partial) transformation ex : V (G) ! V (G) such that ex is the identity mapping on the domain V (G) n fxg, and is called the elementary annihilator of x. Denote by G the set fex : x 2 V (G)g. We de ne the partial Catalan monoid PC (G) of G to be the monoid of transformations of V (G) generated by the elementary transitions and elementary annihilators.

We extend the map
to a map from (G [ G) , the free monoid on letters fs and ex, for s 2 E (G); x 2 V (G), onto PC (G). Notice that this restricts to a map from G  onto C (G).

4 Extensive Local Endomorphisms of Trees In the sequel, unless otherwise stated, all graphs will be acyclic, every transformation will have nite shift and `endomorphism' will be taken to mean `order-endomorphism' or `order-preserving transformation'.

Remark 4.1 Contrast the de nition of an order-endomorphism of an (acyclic) graph with the usual notion of graph-endomorphism : a transformation  of V (G) such that for all x; y 2
(), (x; y) 2 E (G) implies that (x; y) 2 E (G).

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De nition 4.2 A graph whose underlying undirected graph has no loops or cycles is a tree . Notice that, by this de nition, a tree may be disconnected. In this section we give alternative descriptions of the Catalan monoid and partial Catalan monoid of a tree. In order to do this we will require some further notation concerning trees. Let x and y be vertices of a tree T . If x  y then, by the uniqueness of paths in a tree, there is a unique subchain of T of nite length, beginning in x and ending in y, say x = x1 ! x2 !


! xk = y. De ne a subset of V (T ) by ( if x  y [x; y] = f;x1; x2 ; : : : ; xk g otherwise. The same set, without the endpoints is denoted (x; y). The sets (x; y] and [x; y) are de ned in the obvious way. When x  y as above, we are able to de ne the word v(x; y) in T  to be f(x1 ;x2) f(x2 ;x3) : : : f(xk?1 ;xk) if k > 1 or the empty word if k = 1. A vertex in a tree which has in-degree 0 is called a root , and a vertex with out-degree 0 is called a leaf.

De nition 4.3 Let T be any tree and  a transformation of V (T ) with nite shift. For any chain C of T de ne the subset

XC := fx 2
() \ V (C ) j x 2 V (C )g of V (C ). Let C be the restriction of  to XC . Then, of all transformations of V (C ) which are restrictions of , C has the largest domain, and we can call it the localization of  to C . If the localization of  to every chain of T is a (partial) endomorphism, then we say that  is a local endomorphism of T . De ne LE (T ) to be the monoid of all extensive full transformations of V (T ) which are local endomorphisms and let LPE (T ) be the monoid of all extensive full or partial transformations of V (T ) which are local endomorphisms. It is clear that for any tree T the elementary transitions of T are in LE (T ) and the elementary transitions and annihilators are in LPE (T ) so that, C (T )  LE (T ) and PC (T )  LPE (T ). The rest of this section is devoted to showing that these containments are actually equalities.

Remark 4.4 The notation LE (T ) and LPE (T ) may be regarded as extending the

notation of [13], where the monoid of all extensive full transformations of the chain [n] = 1 2


n is denoted En. For any graph G, we denote by E (G) (PE (G)) the monoid of all extensive full (full or partial) transformations of V (G). We now make a de nition which ensures that every element of LPE (T ) can be regarded as a transformation of some nite subtree of T .

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De nition 4.5 Let T be a tree and  2 LPE (T ). The cambium T  of T under  is a tree with vertex set

fx 2 V (T ) j x 2 Sh() or, for some y 2 Sh(); y ! x 2 E (T )g: We de ne the edges of T  to be (V (T )  V (T )) \ E (T ). In a sense which is made precise in the following proposition, the cambium is the subtree of T which is `changed' by .

Proposition 4.6 Let T be a tree and  2 LPE (T ). Then:

(a) T 1 is the empty tree; (b) V (T ) is nite; (c) for each w 2 (T [ T ), w 2 (T w [ T w ). Thus we can regard  quite naturally as an element of LPE (T ).

Proof. Part (a) is immediate and part (b) follows from the fact that Sh() is nite

and each vertex in T has nite out-degree. To see (c), suppose that f(x;y) is a letter of w for some (x; y) 2 E (T ). Clearly x 6= xw so that x 2 Sh(w)  V (T w ). Since (x; y) 2 E (T ), y 2 V (T w ) as well. Suppose ex is a letter of w for some x 2 V (T ). Write w = uexv. If x 2
(w) then xu 6= x so that x 2 Sh(w)  V (T w ). If x 2=
(w) then x 2 Sh(w)  V (T w ) as required.

De nition 4.7 (simple root) Let  2 LPE (T ) and r be a root of T . Then we say that r is a simple root for  if r 2=
() or [r; r) \ Im() = ;. Lemma 4.8 Every  2 LPE (T ) has a simple root in each connected component of T .

Proof. Suppose to the contrary that in some connected component of T , for every root vertex r1 , r1 2
() and there is another vertex x1 such that r1  < x1   r1. If x1 < r1 then r1  < x1   x1 < r1 which contradicts the fact that  is

an endomorphism when localized to a chain containing r1 ; x1; x1 and r1 . Thus x1  r2 for some root r2 6= r1 . By assumption, there is another vertex x2 such that r2 < x2   r2 , which lies under a root r3 6= r2, and so on, yielding a sequence of roots. By niteness of T , there must be some positive integers k and l > 1 such that rk = rk+l, which implies that there is a non-trivial cycle in the underlying undirected graph of T , contradicting the fact that T  is a tree. For example, such a cycle occurs when k = 1 and l = 2, as depicted below. (Arrows indicate paths of 0 or more edges.)

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r1 =...r3

r2

 ...  ... x1+ =...........z.... x?2  x?1 r?1

r?2 

In the sequel, as above, we use the symbols ;  to refer to the natural partial order on V (T ). For another order  on V (T ) we will write x > y for (x; y) 2 . Let  be a total order on V (T ) which extends the natural partial order. Then we say that  is a natural total order on V (T ). Fix a natural total order  on V (T ).

De nition 4.9 (sequence decomposition) Let  2 LPE (T ). Then we de ne the

sequence decomposition Q() of  to be the sequence (i )i, constructed as follows. Let y be the -maximum simple root of T  for . De ne 0 to be the map with domain
() [ fyg, and action given by: ( x=y 0 x = xx ifotherwise. If  is the identity, then Q() is the empty sequence, otherwise Q() is the sequence Q(0) with  appended. That this recursive de nition makes sense is a result of Proposition 4.10 (b) below, which assures us that 0 2 LPE (T ). In general we will write the sequence decomposition of  as (1; : : : ; n) where n = , and we will write the -maximum simple root of T i for i as xi.

For any element  2 LPE (T ) we can now de ne f () 2 (T [ T ) to be the empty word if  = 1, otherwise f () = v1 : : : vn, where ( if xi 2=
() vi = vex(ix ; x ) otherwise. i i

Proposition 4.10 Let  2 C (T ). We have the following facts about the sequence

decomposition of : (a) Sh() = fx1 ; : : : ; xng; (b) for each 1  i  n, i 2 LPE (T ) and the sequence decomposition of i consists of the rst i terms of the sequence decomposition of ; (c) f () = .

Proof. (a) Recall the de nition of T i . Since xi is a root of T i , there is no vertex y such that y ! xi is an edge of T i . Thus xi must be in Sh(i)  Sh(). Conversely, suppose x 2 Sh() = Sh(n). Let i be minimal such that x 2 Sh(i). Now, either x 2=
(i) or xi = 6 x. If x 2=
(i) then, by assumption, x 2
(i?1), whence xi?1 = x and x = xi . Otherwise, xi = 6 x while xi?1 = x. Once again, by the de nition of the sequence decomposition of , x = xi.

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(b) To prove that each element in the sequence decomposition is in LPE (T ), we simply show that if i is then so is i?1. Suppose not, then there is a chain C in T containing xi and some vertex y such that i is order preserving on C but i?1 reverses the order of xi and y. If xi > y then yi?1 > xi = xii?1 and yi?1 = yi > y, which is impossible since i 2 LPE (T ). Otherwise y > xi and yi?1 < xi = xii?1, whence y 2 Sh(i?1)  Sh(i)  V (T i ). Now since xi is a root of a connected component of T i , y must be in a dierent connected component. But xi 2 (y; yi = yi?1) and since T is a tree, [y; yi)  Sh(i). Thus, xi and y belong to the same connected component, which is a contradiction. The second part of (b) is now immediate from the recursive de nition of the sequence decomposition. We prove (c) by induction on n. If n = 0 the result is immediate, so assume n > 0. From (b) we have f (n) = f (n?1)vn so that f (n) = n?1vn, by the inductive hypothesis. We proceed by calculating f (n). First we calculate its domain. If x 2
(n) then certainly x 2
(n?1 ). Were xn?1 2=
(vn ) then xn?1 would be xn, and vn = exn . But since xn is a root of T n , the only vertex which maps to xn under n?1 is xn itself. That x = xn would then contradict the fact that x 2
(n), thus x 2
(f (n)). Conversely, if x 2=
(n) then either x 2=
(n?1) or x = xn and vn = exn . In either case, x 2=
(f (n)). Thus we have shown that
(f (n)) =
(n). Now we show that xn = xf (n) for each x 2
(n). If x = xn, then xn?1v(xn; xn) = xv(xn ; xn) = xn. Suppose, therefore, that x 6= xn, so that xn?1 = xn. If x 2 V (T n ) then xn 2= [xn; xnn) since xn is a simple root of T n for n, whence xf (n) = xnvn = xn. Otherwise x 2= V (T n ) so that xn = x. Certainly xn?1 = x and if x 2 [xn; xnn) then x 2 V (T n ) | a contradiction. Thus, xf (n) = xn?1 vn = xvn = x = xn , as required. Theorem 4.11 PC (T ) = LPE (T ) and C (T ) = LE (T ). In other words, the (partial) Catalan monoid of a tree consists of all extensive (partial) transformations of nite shift which are local endomorphisms. Proof. As noted above, it is clear that PC (T )  LPE (T ) and C (T )  LE (T ). The reverse inclusions are a result of Proposition 4.10 (c) and the fact that f () 2 T  if  2 LE (T ). Remark 4.12 The theorem does not hold if T is not a tree. Consider, by way of example, the graph G with vertices f1; 2; 3g and edges f1 ! 2; 2 ! 3; 1 ! 3g. The generator f(1;3) of C (G) is certainly not an endomorphism of the chain 1 ! 2 ! 3. As foreshadowed in the previous section we have the following corollary, which previously appeared in [1] for the case of full transformations, and [14] for the case of partial transformations. Corollary 4.13 The (partial) Catalan monoid of the graph 1 2


n consists of all extensive (partial) transformations which are endomorphisms.

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5 Presentations of monoids

Let A be an alphabet and let R be a subset of A, referred to as a set of relations on A . The smallest congruence on A generated by R is denoted R], and the monoid A =R], is said to be given by the presentation with generating set A and relations R. We will often write this monoid as A=R or simply hA j Ri. Let u and v be two words in A. If they are identical, then we write u = v. If one may be obtained from the other by a nite number of applications of the relations R, then we write u  v, and say that they belong to the same congruence class of A with respect to R] . Denote the congruence class containing u by [u]. Then the element of A=R] represented by u is [u] and u  v by the relations R if and only if [u] = [v] in A=R] . De nition 5.1 Let f : A ! A be a function such that for all u; v 2 A , f (u)  u and u  v implies f (u) = f (v). Then we say that f is a canonical form function for A =R], and f (u) is the canonical form of u. We will make use of the following well-known general facts about presentations without comment. The proofs are routine, and are omitted. Theorem 5.2 Let A and B be disjoint alphabets with R  A  A and S  B   B  . Then a hA j Ri hB j S i = hA [ B j R; S i: Note that in the theorem above, ` denotes the coproduct in the category of monoids which is the same as the free product of monoids.

Theorem 5.3 Let A be an alphabet and let R and S be subsets of A  A . Then hA j R; S i = hA j Ri=Sb where Sb = f([w1]; [w2]) j (w1; w2) 2 S g, and [w1]; [w2] 2 hA j Ri.

6 Presentations for the Catalan monoids of trees Fix some tree T .

Proposition 6.1 The submonoid of PC (T ) generated by the elementary annihilators is isomorphic to the semilattice of subsets of V (T ) under intersection which has presentation P(T ) = hex 2 T j e2x = ex (P1); ex ey = ey ex (P2)i where ex represents the subset V (T ) n fxg =
(ex). 2

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If we have a presentation for PC (T ) in generators fs 2 T and ex 2 T then it is easy to see that for any vertices a, b, c and d of T the following relations must hold: PC1 PC2 PC3 PC4 C1 C2 C3 C4

f(a;b) ea ea f(a;b) f(a;b) eb ecf(a;b) f(a;b) f(b;c)f(a;b) f(a;b) f(a;c) f(a;b) f(c;d) f(a;c) f(b;c)

= = = = = = = =

f(a;b) ea ea eb f(a;b) ec if c 2= fa; bg. f(b;c) f(a;b) f(b;c) = f(a;b) f(b;c) f(a;b) f(c;d) f(a;b) if fa; bg \ fc; dg = ; f(b;c) f(a;c)

Finally, if a1 , a2 , b1 , b2 , c1 , c2 2 V (T ) with a1 > c1 , b1 > c1, c1  c2, c2 > a2, c2 > b2 and [a1; a2 ] \ [b1 ; b2] = [c1 ; c2] then we have the relation scheme: C5

v(c1 ; t)v(a1; a2)v(b1 ; b2) = v(c1; t)v(b1 ; b2)v(a1; a2 ) for every t 2 (c2; a2].

Denote by C the set of all relations in C1{C5, de ne the sets PC and P similarly. De ne the monoid PC(T ) to be the quotient of the free monoid (T [ T ) by the congruence (P [ C [ PC)], and de ne C(T ) to be the quotient of T  by C]. Theorems 5.2 and 5.3 then imply that a d: PC(T )  = (P(T ) C(T ))=PC From these observations we have Theorem 6.2 The map ?C : C(T ) ! C (T ) de ned by [w] 7! w for each w 2 T  is a homomorphism. Similarly, ?PC : PC(T ) ! PC (T ) de ned by [w] 7! w for each w 2 (T [ T ) is a homomorphism.2

Remark 6.3 Notice that the submonoid of PC(T ) generated by T is a homomorphic image of C(T ) as extra relations between words of T  may result from P and PC. So if [u] = [v] as elements of C(T ) then [u] = [v] as elements of PC(T ).

In the sequel we prove that the homomorphisms of Theorem 6.2 are isomorphisms.

Lemma 6.4 Suppose that x ! y is an edge in T , u 2 (T [ T ) and x 2= Im(u). Then [u] = [uf(x;y)] = [uex], as elements of PC(T ). Further, if u 2 T  then [u] = [uf(x;y)] in C(T ).

Proof. We leave the reader to notice that in the following proof we only use relations from C to prove that u  uf(x;y), when u 2 T  . Let u be such that x 2= Im(u). Certainly u = 6 1. If u is a letter, then either u = ex or u = f(x;z) for some z = 6 x. If u = ex then u  uex by P1 and u  uf(x;y) by PC2.

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Otherwise u = f(x;z) so that uf(x;y)  u by C2 and uex = f(x;z)ex  f(x;z) by PC1. This starts an induction on the length of u. Suppose that u = u0f(a;b) for some edge a ! b and u0 6= 1. There are only four possible ways in which the distinct edges a ! b and x ! y can be related in the tree T: (1) y x = a ! b; (2) a ! b = x ! y; (3) a ! b = y x or fa; bg \ fx; yg = ;; (4) x ! y = a ! b. In Case (1), uf(x;y) = u0f(a;b) f(a;y)  u0f(a;b) = u, by C2, while uex = u0f(a;b) ea  = u by PC1. For Cases (2) to (4) we may assume that a 6= x, hence x 2= Im(u0), for otherwise x 2 Im(u). In Case (2), a 2= Im(u0) for otherwise again x 2 Im(u); so, by successive applications of an inductive hypothesis, uf(x;y) = u0f(a;b) f(x;y)  u0f(x;y)  u0  u0f(a;b) = u. Similarly, uex = u0f(a;b) ex  u0ex  u0  u0f(a;b) = u. In Case (3) uf(x;y) = u0f(a;b) f(x;y)  u0f(x;y)f(a;b)  u0f(a;b) = u, by C4, or C3 if fa; bg \ fx; yg = ;. We use PC4 and the inductive hypothesis to deduce that uex = u0f(a;b) ex  u0exf(a;b)  u0f(a;b) = u. Finally, in Case (4), C1 and the inductive hypothesis give

u0f(a;b)

uf(x;y) = u0f(y;b) f(x;y)  u0f(x;y) f(y;b) f(x;y)  u0f(x;y) f(y;b)  u0f(y;b) = u: That uex  u in Case (4) follows in the same manner as for Case (3). We can now assume that u = u0ea for some a 2 V (T ) and u0 6= 1. There are only three possible ways in which the vertex a and edge x ! y can be related in the tree T: (1) a = x ! y; (2) x ! y = a; (3) a 2= fx; yg: In Case (1) uf(x;y) = u0exf(x;y)  u by PC2 while uex  u by P1. In Cases (2) and (3) we may assume that x 2= Im(u0) for otherwise x 2 Im(u). Thus, in Case (2)

uf(x;y) = u0ea f(x;a)  u0f(x;a) ea f(x;a) (by the inductive hypothesis)  u0ea exf(x;a) (by PC3 and P2)  u0ea ex (by PC2)  u0ea = u by P2 and the inductive hypothesis. For the same reasons we also have uex = u0ea ex  u0exea  u0ea = u. In Case (3), PC4 and the inductive hypothesis give uf(x;y) =

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u0ea f(x;y)  u0f(x;y) ea  u0ea = u and by P2, uex  u as in Case (2). The lemma now follows by induction.

Lemma 6.5 Let x1 ! x2 !


! xk be a chain in some tree T . Then for all pairs i  j 2 f1; : : : ; kg we have [v(xi ; xj )v(x1 ; xk )] = [v(x1; xk )v(xi; xj )] = [v(x1 ; xk )] as elements of C(T ).

Proof. We proceed by induction on j ? i in order to prove that v(xi; xj )v(x1; xk )  v(x1; xk ): If j = i then v(xi; xj ) = 1 and we are nished, so assume that j ? i > 0. If j = 1 then j = i, so j = 1 cannot occur. If j = 2 then v(xi; xj ) = f(x1 ;x2) so that v(xi; xj )v(x1; xk ) = f(x1 ;x2) v(x1; xk )  v(x1; xk ) by use of relation C2. Otherwise, j > 2. Use relation C3 to commute f(xj?1 ;xj ) from the end of v(xi; xj ) as far along v(x1 ; xk ) as possible. It will then encounter f(xj?2 ;xj?1) f(xj?1 ;xj ) and by use of relation C1 we can delete it. Thus v (xi ; xj )v (x1 ; xk )  v (xi ; xj ?1 )v (x1 ; xk ) and by appeal to the inductive hypothesis we are done. That [v(x1; xk )v(xi; xj )] = [v(x1 ; xk )] is a direct consequence of Lemma 6.4.

Lemma 6.6 Let x; y; z 2 V (T ). If x  z and y  z then [v(x; z)v(y; z)] = [v(y; z)v(x; z)] as elements of C(T ).

Proof. Let z0  z be the maximal element of [x; z] \ [y; z]. Then v(x; z)v(y; z) = v(x; z0 )v(z0; z)v(y; z)  v(x; z0 )v(y; z) (by Lemma 6.5) = v(x; z0 )v(y; z0)v(z0 ; z)  v(y; z0)v(x; z0 )v(z0 ; z) (by C3 and C4)  v(y; z)v(x; z) as required.

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C(T ) of T

is a presentation for the Catalan monoid

In this section u  v will be taken to mean [u] = [v] as elements of C(T ). Lemma 7.1 Let ; 2 C (T ), and x the -maximum simple root of T  for . Write f ( ) = v(z1; z1 ) : : : v(zm ; zm ). Let W be the triple (u; y; i) where u 2 T , y 2 [x; x] and i 2 f1; : : : ; mg. Let

(W ) = uv(x; y)v(zi; zi ) : : : v(zm; zm ) 2 T :

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De ne W = uv(x; y) and for each j with i  j  m,

Xj = Im( W ) \ (zj ; zj ): If (W ) = , x 2= Sh(u) [ Sh( ) and for each j 2 fi; : : : ; mg, Xj  fzi ; : : : ; zj?1g then (W )  w0 = u0v(x; y0) for some y0  y and u0 such that x 2= Sh(u0).

Proof. We proceed by induction on m ? i. If m ? i = 0, then the result is immediate, so assume that m > i. Suppose [x; y] \ [zi; zi ] = ; then relation C3 gives v(x; y)v(zi; zi )  v(zi; zi )v(x; y). Set W 0 = (uv(zi; zi ); y; i + 1) then (W )  (W 0) = uv(zi; zi )v(x; y)v(zi+1; zi+1 ) : : : v(zm; zm );

W 0 = uv(zi; zi )v(x; y) and for each j , i + 1  j  m, Xj0 = Im( W 0 ) \ (zj ; zj ): We show that W 0 satis es the conditions of the inductive hypothesis. Let t 2 Xj0 , and suppose by way of contradiction that t 2= fzi+1 ; : : : ; zj?1g. If t 2 Xj then t = zi . But zi 2= Im( W 0 ), so t 2 fzi+1; : : : ; zj?1g. Otherwise t 2= Xj so that t 2 Im( W 0 ) n Im( W ), which implies that t = zi . But that zi 2 (zj ; zj ) contradicts the de nition of f ( ) which states that zj is a simple root of T j for j . Thus in either case we have that Xj0  fzi+1; : : : ; zj?1g, and we can appeal to the inductive hypothesis to give the result. Write the chain between x and x as x = x1 ! x2 !


! xn = x so that y = xr for some r 2 [n]. We can now assume that [x; y] \ [zi ; zi ] = [xp; xq ] with p  q  r, so we have the following diagram. ... xn zi ......... .......* . . . ......... . ......... .. y .* . . . . . . . ...j . -xq...... .......xp............................................... .......* . . ......... . . . . . . . . .....

......... ......... ......... j.z

x1 i Suppose that zi 2= Im( W ) then Lemma 6.4 and the fact that Xi is necessarily empty allow us to delete v(zi; zi ), giving (W )  (W 0) where W 0 = (u; y; i + 1). In this case W 0 = W , and Xj0 = Xj for each i + 1  j  m. Suppose that t 2 Xj0 and t 2= fzi+1; : : : ; zj?1g then t = zi 2= Im( W ) = Im( W 0 ), which is a contradiction. Thus, we may appeal to the inductive hypothesis with W 0 to give the result. We can now assume that zi 2 Im( W ), so for some z  zi , we can write z W = zi , and zi = z. Note also that x1 > xp, for if x1 = xp then zi > x1 contradicting the fact that x1 is a root of T . There are, however, several other cases where the paths between points in the diagram above are trivial. We deal with each one in turn. (1) First we deal with the case that zi = xp. In this case zi = xp = xq = y, for otherwise zi 2= Im( W ). Thus we can write (W ) = (W 0) = uv(x; y)v(y; zi )v(zi+1; zi+1 ) : : : (zm ; zm );

13

III-7

where W 0 = (u; zi ; i +1). We only need to show that Xj0  fzi+1; : : : ; zj?1g. Suppose to the contrary that t 2 Xj0 n fzi+1 ; : : : ; zj?1g. Since Xj0  (Xj n fyg) [ fzi g, t must be zi . But zi 2 (zj ; zj ) contradicts the de nition of f ( ). Thus Xj0  fzi+1; : : : ; zj?1g, and we are done by appeal to the inductive hypothesis with W 0. In the sequel we may assume that zi > xp. (2) Consider the case where xq = xr = y. Then

(W ) = uv(x; y)v(zi; y)v(y; zi ) : : : v(zm; zm )  uv(zi; y)v(x; y)v(y; zi ) : : : v(zm; zm ) (by Lemma 6.6) = uv(zi; y)v(x; zi ) : : : v(zm ; zm ) = (W 0) where W 0 = (uv(zi; y); zi ; i + 1). We only need to check that t 2 Xj0 implies that t 2 fzi+1; : : : ; zj?1g. Suppose not, then since Xj0  (Xj n fzi; yg) [ fzi g, t = zi . (Notice that our argument does not depend on y 6= zi .) As above, this yields a contradiction by the de nition of f ( ), and we are able to appeal to the inductive hypothesis. Assume now that zi > xp, x1 > xp and xq > y. (3) Suppose that xq = zi . Now, by the observation above, zi = z for some z  zi. But zi = xq > y  x1 . Thus we have x1 > z > x1  contradicting the fact that x1 is a simple root of T  for . Finally, we are able to assume that x > xp, zi > xp, xq > y, and xq > zi .

(W ) = uv(x; y)v(zi; zi ) : : : v(zm; zm )  uv(xp; y)v(x; y)v(zi; zi ) : : : v(zm ; zm ) (by Lemma 6.5)  uv(xp; y)v(zi; zi )v(x; y)v(zi+1; zi+1 ) : : : v(zm; zm ) (by C5) = (W 0) where W 0 = (uv(xp; y)v(zi; zi ); y; i + 1). As above we prove that t 2 Xj0 implies that t 2 fzi+1; : : : ; zj?1g. Suppose not, then since Xj0  (Xj n fzi; xpg) [ fzi g, t = zi . As above, this yields a contradiction by the de nition of f ( ). By appeal to the inductive hypothesis, the lemma is proven.

Theorem 7.2 The function F : T  ! T  de ned by F (w) = f (w) is a canonical

form function for C(T ).

Proof. Since w1  w2 implies that w1 = w2, it is certainly true that F (w1 ) = F (w2), so it is only necessary to prove that F (w)  w. Let  = w. We go by induction on n = jSh()j. If n = 0 then w = 1 = f () as required, so assume that n > 0, and w = 6 1. In order to show that w  f () = v(x1; x1 ) : : : v(xn; xn), it suces to prove that w  w0v(xn; xn), with w0 = n?1, for then we are able to proceed by induction to give w  w0v(xn; xn)  f (n?1)v(xn; xn) = f ().

14

III-8

Let the chain between xn and xn  be xn = y1 ! y2 !


! yk = xn . Write w = w1v(xn; y)w2 where jw1j is maximal containing no letter fxn ;z , and jv(xn; y)j is maximal. Then w = w1v(xn; y)w2  w1v(xn; y)w20 where w20 is w2 with all letters fxn;z deleted, by repeated application of Lemma 6.4, and the fact that no pre x of w containing v(xn; y) maps a vertex into xn , since it is a root of T . Let = w20 . By the fact that w20 contains no letter fxn;z , Sh( )  (Sh() n fxng) and we can apply the inductive hypothesis to conclude that w20  f ( ). Application of Lemma 7.1 to , and W = (w1; y; 1) gives w  w10 v(xn ; y0) with xn 2= Sh(w10 ). Notice that this implies that y0 = xn. Lemma 6.5 then gives w  w10 v(y2; xn)v(xn; xn). Set w0 = w10 v(y2; xn). It is clear that w0 = n?1 and the theorem is proven. Theorem 7.3 For any tree T , C (T ) is isomorphic to C(T ). That is to say that the generators fs 2 T with the relations C1{C5 form a presentation for C (T ). Proof. Let T be any tree and  any natural total order on V (T ). By Theorem 6.2, there is an epimorphism ?C : C(T ) ! C (T ) de ned by [w] 7! w. Further, this epimorphism is injective, because if w1; w2 2 T  are such that w1 = w2 then by Lemma 7.2 w1  F (w1) = F (w2)  w2. Hence the homomorphism ?C of Theorem 6.2 is an isomorphism, and the theorem is proved. Corollary 7.4 C(T ) is a submonoid of PC(T ), so that ?C is just the restriction of ?PC to C(T ). Proof. Let S be the submonoid of PC(T ) generated by T . Since ?PC : PC(T ) ! PC (T ) maps S onto C (T )  PC (T ), and by Remark 6.3, we can factor ?C : C(T ) ! C (T ) through S . But ?C : C(T ) ! C (T ) was shown to be an isomorphism so we must have that S  = C(T ).

8

is a presentation for the partial Catalan monoid of T

PC(T )

Henceforth, u  v shall mean [u] = [v] in PC(T ).

Lemma 8.1 For every u 2 T ; ex 2 T , uex  u0 for some  2 T ; u0 2 T  such that ju0j  juj. Proof. We show this by induction on juj. If juj = 0 the result is immediate, beginning the induction. Otherwise we may write u = vf(a;b) . Now there are three cases to consider: x 2= fa; bg; x = a; or x = b. In the rst case, uex = vf(a;b) ex  vexf(a;b) by PC4 and we are nished by induction. If x = a then uex = vf(a;b) ea  vf(a;b) = u by PC1. Finally, if x = b then uex = vf(a;b) eb  vea eb by PC3, and we are done by two applications of the inductive hypothesis.

15

III-8

Lemma 8.2 If w 2 (T [ T ), then w  v for some v 2 T  and  = Qx=2
(w) ex. Proof. By repeated application of Lemma 8.1 we have that every word w 2 (T [ T ) there is a word v, with  2 T ; v 2 T  which represents the same element of PC(T ).

Since the pre x  determines the domain of w, and by the fact (Proposition 6.1) that P(T ) is a presentation for the semilattice of subsets of V (T ) under intersection, we are able to transform  into , the required word.

Lemma 8.3 Let w = u with  as in Lemma 8.2,  = w, = u and u 2 T  in the

canonical form of Theorem 7.2 (i.e. u = F (u)). Write u = v(x1; x1 ) : : : v(xk ; xk ) and let xi1 ; : : : ; xim be the subsequence of x1 ; : : : ; xk which are in
(). Then

w  v(xi1 ; xi1 ) : : : v(xim ; xim ) = v(xi1 ; xi1 ) : : : v(xim ; xim ):

Proof. Let the subsequence of x1 ; : : : ; xk which is the complement of xi ; : : : ; xim be written xj ; : : : ; xjn . Notice that fxjp j p 2 [n]g = Sh( ) n
(). We prove the lemma 1

1

by induction on n. If n = 0 then w is already the word required. Otherwise, let x = xjn and write w = u1 v(x; x )u2. Since, by de nition of the canonical form, x 2= V (T jn?1 ), relations P1, P2 and PC4 ensure that we can commute the letter ex from the front, to write w  u1 exv(x; x )u2. If we write the chain between x and x as x ! y1 ! y2 !


! yq = x then

u1exv(x; x )u2 = u1exf(x;y1 ) : : : f(yq?1 ;yq ) u2  u1exf(y1 ;y2) : : : f(yq?1 ;yq )u2 (by PC2)  u1u2 by Lemma 6.4 and the fact that x is a simple root for jn on T jn . By Proposition 4.10 (b) u1 = F (u1), so that u1 satis es the conditions for the inductive hypothesis and the lemma is proven.

De nition 8.4 Say that a word in the form v(xi ; xi ) : : : v(xim ; xim ) as described 1

1

in Lemma 8.3 is in domain simpli ed form. Lemmas 8.2 and 8.3 combine to ensure that every word in (T [ T ) can be written in domain simpli ed form.

Theorem 8.5 If T is any tree, then PC (T ) = PC(T ). Proof. Since we have already established that ?PC : PC(T ) ! PC (T ) is a homomorphism, we only need to show that it is injective. Assume therefore that w1 and w2 are two words in domain simpli ed form such that w1 = w2 =  2 PC (T ).

16

III-8

Lemma 8.6 (inductive proof of Theorem 8.5) Let  2 PC (T ). Suppose u1 = v(x1; x1 ) : : : v(xn; xn ) and

u2 = v(x1 ; x1 ) : : : v(xn ; xn ) for some  2 Symn such that : (1)  = u1 = u2; (2) the distinct elements x1 ; : : : ; xn constitute the set Sh() \
(); (3) if i < j then xi 6> xj and xi 6> xj in the natural partial order on V (T ); (4) for each i in f1; : : : ; j ? 1g, xi  2= [xj ; xj ) and xi  2= [xj ; xj ). Then u1  u2. Since w1 and w2 are in domain simpli ed form, we can write w1 = u1 and w2 = u2, satisfying the preconditions of the lemma. Thus it is sucient to prove the lemma. De ne a total order  on fx1; : : : ; xng by xi  i +1 then u1 = u2, which starts an induction on the size of the set f(i; j ) j i < j and i > jg. Otherwise, let i be the smallest element such that i > i + 1. Notice that neither xi > xi+1 nor xi+1 > xi for otherwise either u1 or u2 would contradict assumption (3). If [xi ; xi ] \ [xi+1 ; xi+1 ] = ;, then relation C3 ensures that u2  u02 where we de ne u02 = v(x1 ; x1 ) : : : v(xi?1 ; xi?1 )v(xi+1 ; xi+1 )v(xi ; xi ) : : : v(xn ; xn ): We may now appeal to the inductive hypothesis with u1 and u02 to give the result. We can now assume that [xi ; xi ] \ [xi+1 ; xi+1 ] = [z1 ; z2] 6= ;. By assumption (2) xi 6= xi+1 , so that xi > z1 and xi+1 > z1 . If xi+1  < z2 then xi  6= z2 by assumption (4). If xi  < z2 then xi+1  6= z2 for otherwise xi+1  2 [xi ; xi ) while xi > xi+1 , contradicting assumption (4) with regard to u1. Thus there are two possible situations:

17

III-9

(a)

....xi  .* . . . . . . . . . ......... ......... . ......... . . . . . . . ...j ..... ......z .............................................. z . 1 2 . * ......... . . . . ..... ......... . . . . . . . . . ......... . . . . . . . . ...j ...  .... xi+1

xi......

xi+1 

(b)

xi......

......... ......... ...j ......z .............................................. -z2 = xi  = xi+1  . 1 . * . . . . . . . . . ......... . . . . . . . . x ... i+1

In Case (b), Lemma 6.6 ensures that u2  u02 as above, and once again we appeal to the inductive hypothesis. In Case (a) transform the subword v(xi ; xi )v(xi+1 ; xi+1 ) of u2 as follows. Lemma 6.5 gives

v(xi ; xi )v(xi+1 ; xi+1 )  v(z1 ; xi )v(xi ; xi )v(xi+1 ; xi+1 )  v(z1 ; xi )v(xi+1 ; xi+1 )v(xi ; xi ) by C5. We prove that we can delete the subword u0 = v(z1 ; xi ) by induction (*) on its length. Let the pre x of u2 preceeding u0 be u00. If z1 2 Im(u00) then since z1 < xi and z1 < xi+1 , z1 = xj  for some j < i. But then the fact that z1 2 [xi ; xi ) contradicts assumption (4). Thus z1 2= Im(u00). By Lemma 6.4 we may delete the rst letter of u0, and by appeal to (*) we can delete u0. Once again, we have shown that u2 can be transformed into u02 . By appeal to the inductive hypothesis the lemma, and hence the theorem, is proven. All the relations apart from C5 were easy to nd as they involve only two generators. Relation C5 was discovered with the assistance of Magma [2] to count the elements of C (T ) and Walker's presentation enumerator [19] to enumerate C(T ).

9 A review of existing work on Catalan monoids Theorem 1 of Howie [8] states that Theorem 9.1 If X is a nite set, then the subsemigroup of TX generated by the idempotents of non-zero defect is TX n GX . In fact, every element of TX n GX is a product of idempotents of defect 1.

18

III-9

Here TX is the full transformation semigroup on the set X , and GX is its subgroup of non-singular transformations. Identifying every idempotent of defect 1 as an elementary transition of the complete graph on X , and since every nite semigroup embeds in TX n GX for some set X , we are able to deduce from Theorem 9.1 that every nite semigroup embeds in the Catalan monoid of some nite graph. More speci c results are obtained in [16] where it is shown that the pseudovariety of nite R-trivial monoids is generated by all C (G) where G is a nite acyclic graph. It is also shown that the pseudovariety of nite J -trivial monoids is generated by all C (G) where G is a nite acyclic graph in which no vertex has out-degree greater than 1. It is shown in [1] and [17] that when Gn = 1 $ 2 $


$ n, the symmetric chain on n vertices, C (Gn) is the monoid of all endomorphisms of the set f1; : : : ; ng of vertices under the usual total ordering. We shall give a presentation for C (Gn) via the presentations for the Catalan monoids of the two subtrees Tn+ = 1 ! 2 !


! n and Tn? = 1 2


n of Gn. Let fi label the generator of C (Tn?) which maps i + 1 to i and xes the other points, and let gi label the generator of C (Tn+) which maps i to i + 1, and xes the other points. Let ? = ffi : 1  i  n ? 1g and + = fgi : 1  i  n ? 1g. With this notation, by a direct transcription of the presentation for the Catalan monoid of a tree given above,

C (Tn?) = C(Tn?) = h? j fi2 = fi;

fi fi+1fi = fi+1 fifi+1 = fi+1fi; fi fj = fj fi where ji ? j j > 1i

and

C (Tn+) = C(Tn+) = h+ j gi2 = gi;

gigi+1gi = gi+1gigi+1 = gigi+1; gigj = gj gi where ji ? j j > 1i:

Now since the chain Tn+ is identical to the chain Tn? (it is just the mirror image), C (Tn+) must be isomorphic to C (Tn?). On the other hand, inspection of the presentations above yields the fact that C(Tn?) is dual to C(Tn+) since we can move from one presentation to the other by simply reversing the words in the relations. Thus we see immediately the result of Higgins [6] that Corollary 9.2 C (Tn?), which is the monoid of non-increasing endomorphisms of the total order 1  2 


 n, is self-dual.

19

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A presentation for C (Gn) in generators ? [ + will satisfy the following relations, which we shall refer to collectively as O. O1 g i fi = fi O2 f i gi = gi O3 fi gj = gj fi if i 2= fj; j ? 1g O4 gi+1 fi = gi+1 O5 fi gi+1 = fi In [1] Azenstat proves that Theorem 9.3 The monoid of all (full) endomorphisms of the total order 1  2  ` ? + b


 n is C (Gn) and has presentation C(Tn ) C(Tn )=O which we denote by On. The following theorem appears in [6]. Denoting the dual of a monoid M by M op , we have that Theorem 9.4 There are embeddings On ,! On+1 op and Oopn ,! On+1 . Where the embedding is the homomorphic extension of the map taking fi 7! fn?i and gi 7! gn+1?i. It is observed in [6] that Corollary 9.2 implies that the pseudovariety J of nite J {trivial monoids, which is generated by the semigroups C (Tn?) for n  1, is self-dual { that is, for each monoid M in J, M op is also in J. It is also observed that Theorem 9.4 implies that the pseudovariety O, generated by On; n  1 is self-dual. With notation as` above, Theorem 8.5 implies that PC (Tn?) has a presentation consisting of P(T ?) C(T ?)=Tb where T refers to the following relations. n

T1 T2 T3 T4

n

fiei+1 ei+1fi fiei eifj

= = = =

fi ei+1 ei+1 ei fj ei if i 2= fj; j + 1g

` Since P(Tn?)  = P(Tn+) we know that PC (Tn+) has presentation P(Tn?) C(Tn+)=Tc0 where T0 will refer to: T10 giei = gi T20 ei gi = ei T30 gi ei+1 = ei ei+1 T40 eigj = gj ei if i 2= fj; j + 1g In [14] and in [17] it is shown that Theorem 9.5 PC (G`n), the monoid of all partial endomorphisms of f1; : : : ; ng is [ T0): isomorphic to P(Tn?) C(Gn)=(T d and in [6] it is shown that the pseudovariety generated by PC (Gn); n  0 is not self{dual.

20

Acknowledgement This work owes much to the author's supervisor, David Easdown (University of Sydney) for his careful proofreading and his contribution to the nal form of Lemma 6.4. Discussions with John M. Howie, Nikola Ruskuc, Peter Higgins and Piergiulio Katis were also of great bene t.

References [1] A. Ja. Azenstat, Generating relations of an endomorphism semigroup of a nite linearly ordered set , Sibir. Mat. Z. 3 (1962), No. 2, 161{169 (Russian). [2] J. Cannon and C. Playoust, An Introduction to Magma , School of Mathematics and Statistics, The University of Sydney, 1994. [3] N. D. Filippov, Graphs and partial groupoids of their directed transformations , Matem. Zapiski Ural. Gos. Univ., 6 (1967), No. 1, 144{156 (Russian). [4] Gracinda M. S. Gomes and John M. Howie, On the ranks of certain semigroups of order{preserving transformations , Semigroup Forum 45 (1992), 272{282. [5] Peter M. Higgins, Combinatorial results for semigroups of order-preserving mappings , Math. Proc. Camb. Phil. Soc.113 (1993), 281{296. [6] Peter M. Higgins, Divisors of semigroups of order preserving mappings on a nite chain , Technical Report 94{4, University of Essex (To appear in Int. J. Algebra and Computation). [7] Peter M. Higgins, \Techniques of Semigroup Theory", Oxford University Press, 1992. [8] John M. Howie, The subsemigroup generated by the idempotents of a full transformation semigroup , J. London Math. Soc. 41 (1966), 707{716. [9] John M. Howie, \An Introduction to Semigroup Theory", Academic Press Inc. (London) Ltd., 1976. [10] John M. Howie, Combinatorial and probabilistic results in transformation semigroups , Words, Languages and Combinatorics 2, 200{206, World Scienti c, Singapore, 1994. [11] S. Mac Lane, \Categories for the Working Mathematician", Springer-Verlag New York Inc., 1971.

21 [12] V. A. Molchanov, Semigroups of Mappings on Graphs , Semigroup Forum, 27 (1983), 155{199. [13] J. -E. Pin, \Varieties of Formal Languages", North Oxford Academic Publishers Ltd., 1986. [14] B. V. Popova, Generating relations of a partial endomorphism semigroup of a nite linear ordered set , Uc. Zap. Leningrad. Gos. Ped. Inst. 238 (1962), 78{88 (Russian). [15] J. Riordan, \Combinatorial Identities", John Wiley and Sons, Inc., 1968. [16] Andrew Solomon, Strati cations of the variety of R-trivial monoids , Research Report 94{40, School of Mathematics and Statistics, The University of Sydney. [17] Andrew Solomon, Monoids of order preserving transformations of a nite chain , Research Report 94{8, School of Mathematics and Statistics, The University of Sydney. [18] A. S. Vernitskii and M. V. Volkov, A proof and a generalization of Higgins' theorem for semigroups of order{preserving mappings , Technical Report, Department of Mathematics and Mechanics, Ural State University 1994. [19] T. G. Walker, Semigroup Enumeration { Computer Implementation and Applications , Ph.D. thesis, University of St Andrews, 1992.