Combinatorial results for certain semigroups of partial isometries of a finite chain

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 58() (014), Pages Combinatorial results for certain semigroups of partial isometries of a finite chain F. Al-Kharousi Department of Mathematics and Statistics Sultan Qaboos University Al-Khod, PC 1 Oman fatma9@squ.edu.om R. Kehinde Department of Mathematics and Statistics Bowen University P. M. B. 84, Iwo, Osun State Nigeria kennyrot000@yahoo.com A. Umar Department of Mathematics and Statistics Sultan Qaboos University Al-Khod, PC 1 Oman aumarh@squ.edu.om Abstract Let I n be the symmetric inverse semigroup on X n = {1,,...,n} under composition of maps and let DP n and ODP n be its subsemigroups of partial isometries and of order-preserving partial isometries of X n, respectively. In this paper we investigate the cardinalities of some equivalences on DP n and ODP n which lead naturally to obtaining the orders of these semigroups. This work was begun when the second named author was visiting Sultan Qaboos University for a month research visit from October to December 010.

2 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Introduction and Preliminaries Let X n = {1,,...,n} and I n be the partial one-to-one transformation semigroup on X n under composition of mappings. Then I n is an inverse semigroup (that is, for all α I n there exists a unique α I n such that α = αα α and α = α αα ). The importance of I n (more commonly known as the symmetric inverse semigroup or monoid) to inverse semigroup theory may be likened to that of the symmetric group S n to group theory. Every finite inverse semigroup S is embeddable in I n,the analogue of Cayley s theorem for finite groups, and to the regular representation of finite semigroups. Thus, just as the study of symmetric, alternating and dihedral groups has made a significant contribution to group theory, so has the study of various subsemigroups of I n ;seeforexample[,4,5,7,15]. A transformation α I n is said to be order-preserving (order-reversing) if ( x, y Dom α) x y = xα yα (xα yα) and, an isometry (or distance-preserving) if ( x, y Dom α) x y = xα yα. We shall denote by DP n and ODP n, the semigroups of partial isometries and of order-preserving partial isometries of an n chain, respectively. Eventhough semigroups of partial isometries on more restrictive but richer mathematical structures have been studied by Wallen [17], and Bracci and Picasso [] the study of corresponding semigroups on chains was only initiated recently by Al-Kharousi et al. [1]. This paper investigates the combinatorial properties of DP n and ODP n, thereby complementing the results in Al-Kharousi et al. [1] which dealt mainly with the algebraic and rank properties of these semigroups. In this section we introduce basic terminologies and quote some elementary results from Section 1 of Al-Kharousi et al. [1] that will be needed in this paper. In Section we obtain the cardinalities of two equivalences defined on DP n and ODP n.these equivalences lead to formulae for the orders of DP n and ODP n as well as new triangles of numbers that were as a result of this work recently recorded in [14]. Lastly, in Section we computed the number of Green s D-classes in DP n. For standard concepts in semigroup and symmetric inverse semigroup theory, see for example [9, 11]. Let DP n = {α I n :( x, y Dom α) x y = xα yα } (1) be the subsemigroup of I n consisting of all partial isometries of X n.alsolet ODP n = {α DP n :( x, y Dom α) x y = xα yα} () be the subsemigroup of DP n consisting of all order-preserving partial isometries of X n. It is clear that if α DP n (α ODP n )thenα 1 DP n (α 1 ODP n )also. Thus we have the following result. Lemma 1.1 DP n and ODP n are inverse subsemigroups of I n. Next, let α be in I n.theheight of α is h(α) = Im α and fix of α is denoted by f(α), and defined by f(α) = F (α), where F (α) ={x Dom α : xα = x}.

3 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Lemma 1. [1, Lemma 1.7] Let α DP n be such that h(α) =p. Then f(α) =0 or 1 or p. Corollary 1. [1, Corollary 1.8] Let α DP n.iff(α) =p>1 then f(α) =h(α). Equivalently, if f(α) > 1 then α is a partial identity. Lemma 1.4 [1, Lemma 1.] Let α DP n. Then α is either order-preserving or order-reversing. Lemma 1.5 [1, Lemma 1.5] Let α DP n.for1 <i<n,iff (α) ={i} then, for all x Dom α, we have x + xα =i. For convenience we shall henceforth denote the set of order-reversing isometries of X n by DP n. Lemma 1.6 [1, Lemma 1.6] Let α DP n. Then x + xα = y + yα for all x, y Dom α. Lemma 1.7 [1, Lemma 1.10] Let α ODP n and f(α) 1. Then α is a partial identity. Lemma 1.8 [1, Lemma 1.11] Let α ODP n. Then α is either strictly orderdecreasing or strictly order-increasing or a partial identity. Combinatorial results Enumerative problems of an essentially combinatorial nature arise naturally in the study of semigroups of transformations. Many numbers and triangle of numbers regarded as combinatorial gems like the Stirling numbers [9, pp. 4 & 96], the factorial [1, 15], the binomial [7], the Fibonacci number [8], Catalan numbers [6], Lah numbers [6, 10], etc., have all featured in these enumeration problems. For a nice survey article concerning combinatorial problems in the symmetric inverse semigroup and some of its subsemigroups we refer the reader to Umar [16]. These enumeration problems lead to many numbers in Sloane s encyclopaedia of integer sequences [14] but there are also others that are not yet or have just been recorded in [14]. As in Umar [16], for natural numbers n p m 0 we define F (n; p) = {α S : h(α) = Im α = p}, () F (n; m) = {α S : f(α) =m} (4) where S is any subsemigroup of I n. Also, let i = a i = a, for all a {p, m}, and 0 i n.

4 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Lemma.1 Let S = ODP n.thenf (n; p 1 )=n and F (n; p n )=1, for all n. Proof. Since all partial injections of height 1 are vacuously partial isometries, the first statement of the lemma follows immediately. For the second statement, it is not difficult to see that there is exactly one order-preserving partial isometry of height 1 n n: (the identity). 1 n Lemma. Let S = ODP n.thenf(n; p )= 1 n(n 1)(n 1), for all n. 6 Proof. First, we say that -subsets of X n (that is, subsets of size ), say A = {a 1,a } and B = {b 1,b }, are of the same type if a 1 a = b 1 b. Now observe that if a 1 a = i (1 i n 1) then there are n i subsets of this type. However, for order-preserving partial isometries, once we choose a -subset as a domain then the possible image sets must be of the same type and there is only one possible orderpreserving bijection between any two -subsets of the same type. It is now clear that F (n; p )=F(n;)= n 1 (n i) = 1 n(n 1)(n 1), as required. 6 Lemma. Let S = ODP n.thenf (n; p) =F (n 1; p 1) + F (n 1; p), for all n p. Proof. (Bijective proof.) Let α ODP n and h(α) = p. Then it is clear that F (n; p) = A + B, wherea = {α ODP n : h(α) =p and n/ Dom α Im α} and B = {α ODP n : h(α) =p and n Dom α Im α}. Define a map θ : {α ODP n 1 : h(α) =p} A by (α)θ = α where xα = xα (x Dom α). This is clearly a bijection since n/ Dom α Im α. Next, define the right (left) waist of α as ϖ + (α) =max(domα) (w + (α) =max(imα))andamapφ:{α ODP n 1 : h(α) = p 1} B by (α)φ = α where (i) xα = xα (x Dom α)andnα = n (if ϖ + (α) =w + (α) ); (ii) xα = xα (x Dom α)andnα = n ϖ + (α)+w + (α) <n (if ϖ + (α) >w + (α)); (iii) x(α ) 1 = xα 1 (x Im α)andn(α ) 1 = n ϖ + (α 1 )+w + (α 1 ) <n (if ϖ + (α) <w + (α)). In all cases h(α )=p, and case (i) coincides with n Dom α Im α ; case (ii) coincides with n Dom α \ Im α ; and case (iii) coincides with n Im α \ Dom α. Thus Φ is onto. Moreover, it is not difficult to see that α in (i) is idempotent; α in (ii) is (strictly) order-decreasing (xα <x); and α in (iii) is (strictly) orderincreasing (xα >x). Thus, Φ is one-to-one. Hence Φ is a bijection, as required. This establishes the statement of the lemma. Proposition.4 Let S = ODP n. Then F (n; p) = (n p+1) n p+1 p,wheren p 1.

5 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Proof. (The proof is by double induction). Basis step: First, note that the formulae for F (n; p 1 ), F (n; p n )andf (n; p ) are true by Lemmas.1 and.. Inductive step: Suppose F (n 1; p) is true for all n 1 p. (This is the induction hypothesis.) Now using Lemma., we see that F (n; p) = F (n 1; p 1) + F (n 1; p) (n p) n 1 (n p 1) n 1 = + (by ind. hyp.) p p 1 p +1 p (n p) p n (n p 1) (n p) n = + p n p p +1 n p (n p)(p +1)+(n p 1)(n p) n = n(p +1) p = (n np + n) n (n p +1) n =, n(p +1) p p +1 p as required. To find the order of ODP n the next lemma seems indispensable. Lemma.5 For any integer n, we have n p=1 n p +1 p +1 n = n n. p Proof. It is enough to observe that n p +1=(n p)+(p +1). Theorem.6 Let S = ODP n. Then ODP n = n (n +1). Proof. This follows from Proposition.4, Lemma.5, and some algebraic manipulation. Lemma.7 Let S = ODP n.thenf(n; m) = n m, for all n m 1. Proof. This follows directly from Lemma 1.5. Proposition.8 Let S = ODP n. Then F (n; m 0 )= n+1 (n +1). Proof. This follows from Theorem.6, Lemma.5 and the fact that ODP n = nm=0 F (n; m).

6 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Remark.9 The triangles of numbers F (n; p) and F (n; m), the sequence F (n; m 0 ) have as a result of this work just been recorded in Sloane [14]. However, ODP n is [14, A09781]. Remark.10 For p = 0, 1 the concepts of order-preserving and order-reversing coincide but distinct otherwise. However, there is a bijection between the two sets for p ; see [5, page, last paragraph]. We now use Remark.10 and Lemma 1.6 to deduce corresponding combinatorial results for DP n from those of ODP n above. Lemma.11 Let S = DP n. Then F (n; p 1 ) = F (n;1) = n and F (n; p n ) = F (n; n) =, for all n. Lemma.1 Let S = DP n.thenf(n; p )=F(n;)= 1 n(n 1)(n 1), for all n. Lemma.1 Let S = DP n.thenf(n; p) =F (n 1; p 1) + F (n 1; p), for all n p. Proposition.14 Let S = DP n. Then F (n; p) = (n p+1) n p+1 p,wheren p. Theorem.15 Let S = DP n. Then DP n = n+1 (n +) 1. Proof. This follows from Proposition.14, Lemma.11 and some algebraic manipulation. Lemma.16 Let S = DP n.thenf(n; m) = n m, for all n m. Proof. This follows from Corollary 1.. Proposition.17 Let S = DP n.thenf (n; m 1 )= (n 1) and F (n 1; m 1 )= ( n 1) + n, for all n 1. Proof. Let F (α) ={i}. Then by Lemma 1.5, for any x Dom α we have x + xα = i. Thus there i 1 possible elements for Dom α :(x, xα) {(1, i 1), (, i ),...,(i 1, 1)}. However, (excluding (i, i)) we see that there are ) i j=0 = ( i j i, possible partial isometries with F (α) ={i}, wherei 1 n (equivalently, i (n+1)/). Moreover, by symmetry we see that F (α) ={i} and F (α) ={n i+1} give rise to equal numbers of partial isometries. Note that if n is odd (even) the equation i = n i + 1 has one (no) solution. Hence, if n =a 1wehave a 1 i + a = (a 1) + a

7 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), partial isometries with exactly one fixed point; if n =a we have a i = (a 1) partial isometries with exactly one fixed point. Proposition.18 Let S = DP n.then F (n; m 0 )= { 1 n (n +9n+10), if n is even; 5 n 1 (n +9n+10), if n is odd. Proof. This follows from Theorem.15, Lemma.16, Proposition.17 and the fact that DP n = n m=0 F (n; m). Remark.19 The triangles of numbers F (n; p) and F (n; m) and, the sequences DP n and F (n; m 0 ), have as a result of this work been recorded recently in Sloane [14]. However, F (n; m 1 ) is [14, A061547]. Number of D-classes First, notice that from [1, Lemma.1] we deduce that number of L-classes in K(n, p) ={α DP n : h(α) =p} (aswellasthenumberofr-classes there) is n p. To count the number of D-classes in DPn and ODP n, first we recall (from [1]) that the gap and reverse gap of the image set of α (with h(α) =p) are ordered (p 1)-tuples defined as follows: g(im α) =( a α a 1 α, a α a α,..., a p α a p 1 α ) and g R (Im α) =( a p α a p 1 α ),..., a α a α, a α a 1 α ), a1 a where α = a p with 1 a a 1 α a α a p α 1 <a < <a p n. Further, let d i = a i+1 α a i α for i =1,,...,p 1. Then For example, if g(im α) =(d 1,d,...,d p 1 )andg R (Im α) =(d p 1,d p,...,d 1 ). α = ,β= DP then g(im α) =(1,,, 1), g(im β) =(,, 1), g R (Im α) =(1,,, 1) and g R (Im β) = (1,, ).

8 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Theorem.1 [1, Theorem.4] Let DP n be as defined in (1) and let α, β DP n. Then (α, β) Dif and only if g(im α) =g(im β) or g R (Im α) =g(im β). Theorem. [1, Theorem.5] Let ODP n be as defined in () and let α, β ODP n.then (α, β) Dif and only if g(im α) =g(im β). Next we prove some preliminary results towards our goal. Lemma. Let p 1 p 1 d i n 1. Proof. The greatest lower bound is attained if d i =1foralli {1,,...,p 1} whilst the least upper bound is attained if α is order-preserving (order-reversing) and a 1 α =1,a p α = n (a 1 α = n, a p α =1). Let d(n, p) be the number of distinct ordered p-tuples: (d 1,d,...,d p ) with p d i = n. This is clearly the number of compositions of n into p parts. Thus, we have Lemma.4 [1, p.151] d(n, p) = n 1 p 1. We shall henceforth use the following well-known binomial identity when needed: n m=p m = p n +1. p +1 An ordered p-tuple: (d 1,d,...,d p )issaidtobesymmetric if (d 1,d,...,d p )=(d 1,d,...,d p ) R =(d p,d p 1,...,d 1 ). Let d s (n, p) be the number of distinct symmetric ordered p-tuples: (d 1,d,...,d p )with p d i = n. Thenwehave 0, if n is odd and p is even; Lemma.5 d s (n; p) = ( n 1 ), otherwise. p 1 Proof. First notice that for symmetric ordered p-tuples the right-half is the reverse of the left-half. Of course if p is odd then the left-half and right-half would exclude the middle term which must be even (odd) if n is even (odd), respectively. Next, observe that if p is even then p/ d i = p/ d p i

9 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), and so p/ p d i = d i = n which implies that n must be even. In other words, if n is odd and p is even then d s (n, p) =0, as required. Otherwise, we have three cases: (i) n and p are both odd. d s (n, p) = d( i 1 n i +1, p 1 )= i 1 (ii) n and p are both even. d s (n, p) =d( n, p ( n ) )= p ; (iii) n is even and p is odd. d s (n, p) = i 1 d( n i, p 1 )= i 1 ( n i 1 ) p = ( n i ) p = ( n 1 ) p 1 ; ( n ) p 1. Define an equivalence R on the class of ordered p-tuples: (d 1,d,...,d p )with p d i = n by (a 1,a,...,a p ) R (b 1,b,...,b p ) if and only if (a 1,a,...,a p )=(b 1,b,...,b p ) or (a 1,a,...,a p )=(b 1,b,...,b p ) R. Let e(n, p) be the number of these equivalence classes. Then we have Lemma.6 e(n; p) = n 1 p 1, if n is odd and p is even; ) ( n 1 ) + ], otherwise. 1 1 [( n 1 p 1 p 1 Proof. The result follows from Lemmas.4 and.5 and the observation that e(n, p) = d(n, p) d s(n, p) Now we have the main result of this section. + d s (n, p) = d(n, p)+d s(n, p). Theorem.7 Let B(n, p) be the number of D-classes of height p in DP n. Then B(n, 0) = 1 and for n p 1 we have ) ( 1 [( n 1 n 1 ) p 1 + p ], if n is odd and p is even; ) ( 1 B(n, p) = [( n 1 n ) ( n ) p 1 + p + p ], if n, p are both even; ) ( n ) + ], otherwise. 1 [( n 1 p 1 p 1

10 F. AL-KHAROUSI ET AL. / AUSTRALAS. J. COMBIN. 58 () (014), Proof. The result follows from Lemma.6 and the fact that B(n, p) = n 1 i=p 1 e(i, p 1). Corollary.8 { n n + n, if n is even; p=0 B(n, p) = n + n, if n is odd. Proof. The result follows from Theorem.7. Remark.9 ThetriangleofnumbersB(n; p) and seqence p 0 B(n, p) have as at the time of submitting this paper not yet been recorded in Sloane [14]. However, d s (n; p) and e(n, p) are [14, A051159] and [14, A04851], respectively. Acknowledgements We wish to thank an anonymous referee for helpful comments and suggestions. The second named author would like to thank Bowen University, Iwo for financial support and Sultan Qaboos University for hospitality. References [1] F. Al-Kharousi, R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain Comm. Algebra (to appear). [] D. Borwein, S. Rankin and L. Renner, Enumeration of injective partial transformations, Discrete Math. 7 (1989), [] L. Bracci and L. E. Picasso, Representations of semigroups of partial isometries, Bull. Lond. Math. Soc. 9 (007), [4] V. H. Fernandes, The monoid of all injective orientation-preserving partial transformations on a finite chain, Comm. Algebra 8 (000), [5] V. H. Fernandes, G. M. S. Gomes and M. M. Jesus, The cardinal and idempotent number of various monoids of transformations on a finite chain, Bull. Malays. Math. Sci. Soc. 4 (011), [6] O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups. An Introduction, Springer-Verlag, London, 009. [7] G. U. Garba, Nilpotents in semigroups of partial one-to-one order-preserving mappings, Semigroup Forum 48 (1994), [8] J. M. Howie, Products of idempotents in certain semigroups of transformations, Proc.EdinburghMath.Soc.17 (1971), 6.

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