Introduction

Let T_n be the full transformation semigroup on the set 5#5, that is, the set of all maps 6#6 under the operation of composition. Associated with each 1#1 in T_n is a digraph 7#7 whose vertices are 1,2,...,n and where (i,j) is a (directed) edge if and only if 8#8. Each connected component 9#9 of 7#7 has a core 10#10 defined by the property that 11#11 is in 10#10 if and only if there exists p > 0 such that 12#12. In the terminology of [2], 9#9 is called standard if 13#13, acyclic if 14#14, cyclic if 15#15, and singleton if 16#16. Thus, for example, if n = 9 and

17#17 (1)

then 7#7 is as shown in Figure 1,

 

Figure: Components of 1#1

18#18

and the components, reading from left to right are respectively standard, acyclic, cyclic, and singleton.

The union of all the cores 10#10 may reasonably be called the core of 1#1 and written 19#19. In fact,

20#20

For every x in X_n the 1#1-height of x is defined as min 21#21.

A group element 1#1 of T_n is one that lies in maximal subgroup of T_n. It is well-known that 1#1 is a group element if and only if the image 22#22 is a transversal of the equivalence 23#23 ( = 24#24. Equivalently, 1#1 is a group element of T_n if and only if 25#25is one-to-one.

The concept of a group element is related to that of 1#1-height by the following result.

Lemma 1.1   An element 1#1 of T_n is a group element if and only if every xin X_n has 1#1-height 0 or 1.

Proof: It is clear that 26#26 and that 27#27 is one-to-one. If every x in X_n has 1#1-height 0 or 1, then 28#28; hence, 25#25 is one-to-one and so 1#1 is a group element.

Conversely, suppose that there exists x in X_n with 1#1-height 29#29. Let x be in a component 9#9 (say), where 30#30. Then

31#31

Thus, 25#25 is not one-to-one and so 1#1 is not a group element. 32#32

The symmetric group S_n is of course contained in T_n, but we shall here be concerned with 33#33, the subsemigroup of T_n consisting of all singular self-maps of X_n. We shall denote this semigroup by Sing_n.

For 1#1 in Sing_n we define 34#34 to be the number of cyclic components in 7#7 and 35#35 to be the number of fixed points of 1#1. In fact 35#35 ( 36#36 ) is the number of acyclic components plus the number of singleton components.

Let E denote the set of idempotents in Sing_n and let E₁be the subset of E consisting of all idempotents 1#1 such that 37#37. It has been known for some time[2] that E, and even E₁, is a set of generators for Sing_n:

38#38

In [5] and [2] the function 39#39 given by

40#40

was shown to be of interest. Following [5] we call it the gravity of 1#1. For each 1#1 in Sing_n the shortest product of idempotents from E₁ that equals 1#1 is of length 3#3. In other words,

41#41

Thus in example 1, where 42#42, we can conclude that 43#43 but 44#44.

Much more recently Tatsuhiko Saito has shown[6] that 3#3 is relevant also to the more general problem of expressing 1#1 as a product of elements of E. Let 45#45 (the defect of 1#1) be defined by 46#46, and let 19#19 be defined by the property that 47#47, 48#48Saito's theorem states that

49#49

moreover 50#50 if 51#51. (Here 52#52 denotes the least integer M such that 53#53.)

In this paper we begin by exploring some arithmetical consequences of Saito's results. Then we examine the function 3#3 more closely, obtaining some formulae in relatively special cases.