Bulletin of the Malaysian Mathematical Sciences Society, 2, 34 (1),(2011), p. 79–85We consider various classes of monoids of transformations on a finite chain, in particular transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested
in computing both their cardinals and the number of their idempotents.
In this note we present a short survey on these questions which have been approached by various authors and close the problem by computing the number of idempotents of those monoids not considered before.
Fibonacci and Lucas numbers play an essential role in the last computations

Fernandes, Vítor H.

Gomes, Gracinda M. S.

Jesus, Manuel M.

Bulletin of the Malaysian Mathematical Sciences Society

English

Repositório da Universidade Nova de Lisboa

DM-FCT-UNL pre-printsThe cardinal and the idempotent number of various monoidsof transformations on a finite chainVı́tor H. Fernandes, Gracinda M. S. Gomes, Manuel M. Jesus 1Centro de Álgebra da Universidade de Lisboa,Av. Prof. Gama Pinto, 2,1649-003 Lisboa,PortugalE-mails: vhf@fct.unl.pt, ggomes@cii.fc.ul.pt, mrj@fct.unl.ptAbstractWe consider various classes of monoids of transformations on a finite chain, in particular transformationsthat preserve or reverse either the order or the orientation. Being finite monoids we are naturally interestedin computing both their cardinals and the number of their idempotents.In this note we present a short survey on these questions which have been approached by various authorsand close the problem by computing the number of idempotents of those monoids not considered before.Fibonacci and Lucas numbers play an essential role in the last computations.2000 Mathematics Subject Classification: Primary 20M17; Secondary 20M20, 05A10.Keywords: monoid, transformation, order-preserving, orientation-preserving.IntroductionLet Xn be a finite chain with n elements, say Xn = {1 < · · · < n}. We denote by PT n the monoid (undercomposition) of all partial transformations of Xn. The submonoids of PT n of all full transformations of Xnand of all injective partial transformations of Xn are denoted by Tn and In, respectively.For general background on monoids, we refer the reader to Howie’s book [9]. Given s ∈ PT n, we denote byDom(s) its domain and by Im(s) its image and define its rank as being the cardinal of Im(s).We say that a transformation s in PT n is order-preserving (resp., order-reversing) if x ≤ y implies xs ≤ ys(resp., xs ≥ ys), for all x, y ∈ Dom(s). Clearly the product of two order-preserving transformations or of twoorder-reversing transformations is order-preserving and the product of an order-preserving transformation byan order-reversing transformation is order-reversing.Denote by POn the submonoid of PT n of all partial order-preserving transformations of Xn. As usual, wedenote by On the monoid POn ∩ Tn of all full transformations of Xn that preserve the order. The cardinaland the number of idempotents of On were calculated by Howie [8]. Later on, Gomes and Howie [7] consideredthese questions for POn. The injective counterpart of On is the inverse monoid POIn = POn ∩ In of allinjective order-preserving partial transformations of Xn. The cardinal of POIn was calculated by Fernandes[2]. Obviously POIn and In have the same idempotents, which are the partial identities on Xn, and there areexactly 2n such elements.1This work was developed within the activities of Centro de Álgebra da Universidade de Lisboa, supported by FCT and FEDER,within project POCTI-ISFL-1-143.1DM-FCT-UNL pre-printsWider classes of monoids are obtained when we take transformations that either preserve or reverse theorder. In this way, we get PODn, the submonoid of PT n of all partial transformations that preserve or reversethe order. Naturally, we may also consider its submonoids ODn = PODn∩Tn and PODIn = PODn∩In. Thecardinals of these monoids were calculated by the authors [4, 5].Now, let a = (a1, a2, . . . , at) be a sequence of t (t ≥ 0) elements from the chain Xn. We say that a is cyclic(resp., anti-cyclic) if there exists no more than one index i ∈ {1, . . . , t} such that ai > ai+1 (resp., ai < ai+1),where at+1 denotes a1. Let s ∈ PT n and suppose that Dom(s) = {a1, . . . , at}, with t ≥ 0 and a1 < · · · < at. Wesay that s is an orientation-preserving (resp., orientation-reversing) transformation if the sequence of its images(a1s, . . . , ats) is cyclic (resp., anti-cyclic). These notions were introduced by McAlister [10] and, independently,by Catarino and Higgins [1]. As before the product of two orientation-preserving or of two orientation-reversingtransformations is orientation-preserving and the product of an orientation-preserving transformation by anorientation-reversing transformation is orientation-reversing.We denote by POPn the submonoid of PT n of all orientation-preserving transformations and by PORnthe submonoid of PT n of all orientation-preserving transformations together with all orientation-reversingtransformations. Denote by OPn the submonoid of POPn of all (orientation-preserving) full transformations,by ORn the submonoid of PORn of all (orientation-preserving or orientation-reversing) full transformations,by POPIn the inverse submonoid of In whose elements are all orientation-preserving transformations and byPORIn the inverse submonoid of In whose elements belong to PORn. The cardinals of OPn and ORn werecalculated by McAlister [10] and, independently, by Catarino and Higgins [1], who also computed the numberof their idempotents. In [3] Fernandes calculated the cardinal of POPIn and the authors, in [4], answered thissame question for PORIn. Again, it is easy to show that the idempotents of PODIn, POPIn and PORInare exactly the 2n idempotents of In.In what follows we denote by 1 the trivial monoid, by Sn the symmetric group and by Cn the cyclic groupof order n. The diagram bellow, with respect to the inclusion relation, clarifies the relationship between thesevarious monoids:•Tn• PTn•In @@@@@@      •      @@@@@@Sn•    @@@@Cn•    @@POPIn •@@@@  OPn•      PORIn •@@@@@@ORn•POPn•PORn•    @@@@1AAA•    @@POInAAA•@@@@  OnAAA•      PODIn AAA•@@@@@@ODnAAA•POnAAA•PODnAAAThe first section of this note is dedicated to recalling known formulas to calculate the cardinal of the monoidsdefined above. In the second section we consider the analogous question for the set of idempotents of thosemonoids. Here we revise the known results and complete the study by computing the remaining cases.1 CardinalsLet PDn be the set of all order-reversing partial transformations of Xn and let IDn and Dn be the subsets ofPDn of all injective transformations and of all full transformations, respectively. Clearly, PODn = POn∪PDnand so PODIn = POIn ∪ IDn and ODn = On ∪ Dn. Furthermore, POn ∩ PDn = {s ∈ PTn : | Im(s)| ≤ 1}and so POIn ∩ IDn = {s ∈ In : | Im(s)| ≤ 1} and On ∩ Dn = {s ∈ Tn : | Im(s)| = 1}.2DM-FCT-UNL pre-printsConsider the following order-reversing permutation of order two:h =(1 2 · · · n− 1 nn n− 1 · · · 2 1).To calculate the cardinals of PODn, PODIn and ODn, we will make use of the mapping ϕ : POn −→ PDndefined by (s)ϕ = sh, for all s ∈ POn. Clearly, this mapping ϕ is a bijection and so we have |PDn| = |POn|.On the other hand, ϕ maps On onto Dn and POIn onto IDn, whence |Dn| = |On| and |IDn| = |POIn|.Now, we recall that Howie [8] showed that|On| =(2n− 1n− 1)and since we have |On ∩ Dn| = |{s ∈ Tn : | Im(s)| = 1}| = n, the next result followed.Theorem 1.1 [5] |ODn| = 2(2n− 1n− 1)− n. As |POIn ∩ IDn| = |{s ∈ In : | Im(s)| ≤ 1}| = n2 + 1 and Fernandes [2] proved that|POIn| =(2nn),we deduced the cardinal of PODIn.Theorem 1.2 [4] |PODIn| = 2(2nn)− n2 − 1. Next, we noticed that Gomes and Howie [7] established that|POn| =n∑i=1(ni)(n + i− 1i)+ 1and using the fact that |POn ∩ PDn| = |{s ∈ PTn : | Im(s)| ≤ 1}| = n∑ni=1(ni)+ 1, we computed |PODn|.Theorem 1.3 [5] |PODn| =n∑i=1(ni) (2(n + i− 1i)− n)+ 1. The cardinal of POPn was also calculated by the authors.Theorem 1.4 [6] |POPn| = 1 + (2n − 1)n +n∑k=2k(nk)22n−k. Denote by Porn the set of all orientation-reversing partial transformations of Xn. By definition, we havePORn = POPn ∪ Porn. Catarino and Higgins proved the following fact:Lemma 1.5 [1] Let a be a cyclic (resp., anti-cyclic) sequence. Then a is also anti-cyclic (resp., cyclic) if andonly if a has no more than two distinct values. This allows us to conclude that POPn ∩ Porn = {s ∈ POPn : | Im(s)| ≤ 2}. As the mapping Ψ :POPn −→ Porn defined by (s)Ψ = sh, for all s ∈ POPn, is a bijection, we get |POPn| = |Porn| and so|PORn| = 2|POPn| − |{s ∈ POPn | | Im(s)| ≤ 2}|. Therefore we obtain the cardinal of PORn.Theorem 1.6 [6] |PORn| = 1 + (2n − 1)n + 2(n2)22n−2 +n∑k=32k(nk)22n−k. 3DM-FCT-UNL pre-printsThe cardinal of POPIn, computed by Fernandes [3], is given by the following formula|POPIn| = 1 +n2(2nn).As, in this case Ψ maps POPIn onto the set of all injective orientation-reversing transformations, we concludethat |PORIn| = 2|POPIn| − |{s ∈ POPIn | | Im(s)| ≤ 2}|. We then proved the following.Theorem 1.7 [4] |PORIn| = 1 + n(2nn)− n22(n2 − 2n + 3). The cardinals of OPn and ORn were computed by McAlister [10] and, independently, by Catarino andHiggins [1], who proved that|OPn| = n(2n− 1n− 1)− n(n− 1) and |ORn| = n(2nn)− n22(n2 − 2n + 5) + n.Just to complete the picture recall that|Cn| = n, |Sn| = n!, |In| =n∑k=0(nk)2k!, |Tn| = nn and |PTn| = (n + 1)n.2 Number of idempotentsFor a given monoid M , we denote by E(M) its set of idempotents.First we will consider the order case. Let M ∈ {ODn,PODIn,PODn}. Let e ∈ E(M). As e2 = e and theproduct of two order-preserving transformations or of two order-reversing transformations is an order-preservingtransformation, we conclude that e must be order-preserving. Thus E(ODn) = E(On) and E(PODn) =E(POn).In [8] Howie showed that|E(On)| = F2n,where Fn is the nth Fibonacci number.Recall that, the Fibonacci numbers are recursively defined byF0 = 0, F1 = 1, Fk+1 = Fk + Fk−1, k ≥ 1.Another interesting sequence of numbers is the Lucas numbers, which are also recursively defined as followsL0 = 2, L1 = 1, Lk+1 = Lk + Lk−1, k ≥ 1.Fibonacci and Lucas numbers are intrinsically related, in fact, for any n ∈ N0,Fn =τn − θnτ − θand Ln = τn + θn,where τ is the golden number and θ is its rational conjugate, that is τ = 1+√52 and θ =1−√52 , and we haveF2n = FnLn. For further details, see e.g. [11].In view of the above observations, we conclude that|E(ODn)| = |E(On)| = F2n =τ2n − θ2nτ − θ.Concerning the correspondent classes of partial transformations, we prove the following.4DM-FCT-UNL pre-printsTheorem 2.1 |E(PODn)| = |E(POn)| =n∑j=1(nj)F2j + 1 =n∑j=1(nj) (τ2j − θ2jτ − θ)+ 1.Proof. First, notice that an element s ∈ PTn is idempotent if and only ifIm(s) ⊆ Fix(s) = {x ∈ Dom(s) | xs = x}.Next, observe that for each nonempty subset A of Xn, the number of idempotents of POn with domain Acoincides with |E(O|A|)|. Putting these facts together, we get|E(PODn)| = |E(POn)| =∑nj=1(nj)|E(Oj)|+ 1 =∑nj=1(nj)F2j + 1 =∑nj=1(nj) (τ2j−θ2jτ−θ)+ 1,as required. We now look into the orientation case. Let M ∈ {ORn,PORIn,PORn}. Let e ∈ E(M). As for the ordercase, the product of two orientation-preserving or of two orientation-reversing elements of M is an orientation-preserving transformation, whence e preserves the orientation. Thus E(ORn) = E(OPn) and E(PORn) =E(POPn).Recall that Catarino and Higgins [1] showed that|E(ORn)| = |E(OPn)| = L2n − (n2 − n + 2) = τ2n + θ2n − (n2 − n + 2).We finish this note by computing the number of idempotents of POPn and of PORn.Theorem 2.2 |E(PORn)|= |E(POPn)|=n∑j=1(nj)[L2j − (j2− j + 2)] + 1=n∑j=1(nj)[τ2j + θ2j − (j2− j + 2)] + 1.Proof. Once again, as an element s ∈ PTn is idempotent if and only if Im(s) ⊆ Fix(s), and for each nonemptysubset A of Xn, the number of idempotents of POPn with domain A coincides with |E(OP |A|)|, we obtain|E(PORn)| = |E(POPn)| =∑nj=1(nj)|E(OPj)|+ 1=∑nj=1(nj)[L2j − (j2 − j + 2)] + 1=∑nj=1(nj) [τ2j + θ2j − (j2 − j + 2)]+ 1,as required. To complete the picture recall that |E(Cn)| = |E(Sn)| = 1, |E(POIn)| = |E(PODIn)| = |E(PORIn)| =|E(In)| = 2n, |E(Tn)| =∑nj=1(nj)jn−j and |E(PTn)| =∑nj=1(nj) (∑jk=1(jk)kj−k)+ 1.References[1] Paula M. Catarino and Peter M. Higgins, The monoid of orientation-preserving mappings on a chain,Semigroup Forum 58 (1999), 190-206.[2] Vı́tor H. Fernandes, Semigroups of order-preserving mappings on a finite chain: a new class of divisors,Semigroup Forum 54 (1997), 230-236.[3] Vı́tor H. Fernandes, The monoid of all injective orientation-preserving partial transformations on a finitechain, Comm. Algebra 28 (2000), 3401-3426.5DM-FCT-UNL pre-prints[4] Vı́tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Presentations for some monoids ofinjective partial transformations on a finite chain, Southeast Asian Bull. Math. 28 (2004), 903-918.[5] Vı́tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Congruences on monoids of order-preserving or order-reversing transformations on a finite chain, Glasgow Math. J. 47 (2005), 413-424.[6] Vı́tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Congruences on monoids of transforma-tions preserving the orientation on a finite chain, submitted.[7] Gracinda M. S. Gomes and John M. Howie, On the ranks of certain semigroups of order-preserving trans-formations, Semigroup Forum 45 (1992), 272-282.[8] John M. Howie, Product of idempotents in certain semigroups of transformations, Proc. Edinburgh Math.Soc. 17 (1971) 223–236.[9] John M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, 1995.[10] Donald McAlister, Semigroups generated by a group and an idempotent, Comm. Algebra 26 (1998), 515-547.[11] Fred S. Roberts, Applied Combinatorics, Prentice-Hall, Inc., Englewood Cliffs, 1984.First and Third authors’ second address: Departamento de Matemática, Faculdade de Ciências e Tecnologia, UniversidadeNova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal.Second author’s second address: Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016Lisboa, Portugal.6

The cardinal and the idempotent number of various monoids of transformations on a finite chain

New University of Lisbon's Repository

DM-FCT-UNL pre-prints
The cardinal and the idempotent number of various monoids
of transformations on a finite chain
Vı´tor H. Fernandes, Gracinda M. S. Gomes, Manuel M. Jesus 1
Centro de A´lgebra da Universidade de Lisboa,
Av. Prof. Gama Pinto, 2,
1649-003 Lisboa,
Portugal
E-mails: vhf@fct.unl.pt, ggomes@cii.fc.ul.pt, mrj@fct.unl.pt
Abstract
We consider various classes of monoids of transformations on a finite chain, in particular transformations
that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested
in computing both their cardinals and the number of their idempotents.
In this note we present a short survey on these questions which have been approached by various authors
and close the problem by computing the number of idempotents of those monoids not considered before.
Fibonacci and Lucas numbers play an essential role in the last computations.
2000 Mathematics Subject Classification: Primary 20M17; Secondary 20M20, 05A10.
Keywords: monoid, transformation, order-preserving, orientation-preserving.
Introduction
Let Xn be a finite chain with n elements, say Xn = {1 < · · · < n}. We denote by PT n the monoid (under
composition) of all partial transformations of Xn. The submonoids of PT n of all full transformations of Xn
and of all injective partial transformations of Xn are denoted by Tn and In, respectively.
For general background on monoids, we refer the reader to Howie’s book [9]. Given s ∈ PT n, we denote by
Dom(s) its domain and by Im(s) its image and define its rank as being the cardinal of Im(s).
We say that a transformation s in PT n is order-preserving (resp., order-reversing) if x ≤ y implies xs ≤ ys
(resp., xs ≥ ys), for all x, y ∈ Dom(s). Clearly the product of two order-preserving transformations or of two
order-reversing transformations is order-preserving and the product of an order-preserving transformation by
an order-reversing transformation is order-reversing.
Denote by POn the submonoid of PT n of all partial order-preserving transformations of Xn. As usual, we
denote by On the monoid POn ∩ Tn of all full transformations of Xn that preserve the order. The cardinal
and the number of idempotents of On were calculated by Howie [8]. Later on, Gomes and Howie [7] considered
these questions for POn. The injective counterpart of On is the inverse monoid POIn = POn ∩ In of all
injective order-preserving partial transformations of Xn. The cardinal of POIn was calculated by Fernandes
[2]. Obviously POIn and In have the same idempotents, which are the partial identities on Xn, and there are
exactly 2n such elements.
1This work was developed within the activities of Centro de A´lgebra da Universidade de Lisboa, supported by FCT and FEDER,
within project POCTI-ISFL-1-143.
1
DM-FCT-UNL pre-prints
Wider classes of monoids are obtained when we take transformations that either preserve or reverse the
order. In this way, we get PODn, the submonoid of PT n of all partial transformations that preserve or reverse
the order. Naturally, we may also consider its submonoids ODn = PODn∩Tn and PODIn = PODn∩In. The
cardinals of these monoids were calculated by the authors [4, 5].
Now, let a = (a1, a2, . . . , at) be a sequence of t (t ≥ 0) elements from the chain Xn. We say that a is cyclic
(resp., anti-cyclic) if there exists no more than one index i ∈ {1, . . . , t} such that ai > ai+1 (resp., ai < ai+1),
where at+1 denotes a1. Let s ∈ PT n and suppose that Dom(s) = {a1, . . . , at}, with t ≥ 0 and a1 < · · · < at. We
say that s is an orientation-preserving (resp., orientation-reversing) transformation if the sequence of its images
(a1s, . . . , ats) is cyclic (resp., anti-cyclic). These notions were introduced by McAlister [10] and, independently,
by Catarino and Higgins [1]. As before the product of two orientation-preserving or of two orientation-reversing
transformations is orientation-preserving and the product of an orientation-preserving transformation by an
orientation-reversing transformation is orientation-reversing.
We denote by POPn the submonoid of PT n of all orientation-preserving transformations and by PORn
the submonoid of PT n of all orientation-preserving transformations together with all orientation-reversing
transformations. Denote by OPn the submonoid of POPn of all (orientation-preserving) full transformations,
by ORn the submonoid of PORn of all (orientation-preserving or orientation-reversing) full transformations,
by POPIn the inverse submonoid of In whose elements are all orientation-preserving transformations and by
PORIn the inverse submonoid of In whose elements belong to PORn. The cardinals of OPn and ORn were
calculated by McAlister [10] and, independently, by Catarino and Higgins [1], who also computed the number
of their idempotents. In [3] Fernandes calculated the cardinal of POPIn and the authors, in [4], answered this
same question for PORIn. Again, it is easy to show that the idempotents of PODIn, POPIn and PORIn
are exactly the 2n idempotents of In.
In what follows we denote by 1 the trivial monoid, by Sn the symmetric group and by Cn the cyclic group
of order n. The diagram bellow, with respect to the inclusion relation, clarifies the relationship between these
various monoids:
•Tn
• PTn
•In @
@
@
@
@
@
 
 
 
 
 
 
• 
 
 
 
 
 
@
@
@
@
@
@
Sn
• 
 
  
@
@
@@
Cn
• 
 
  
@
@
POPIn •@
@
@@
 
 
OPn
• 
 
 
 
 
 
PORIn •@
@
@
@
@
@
ORn
•POPn
•PORn
• 
 
  
@
@
@@
1A
AA
• 
 
  
@
@
POInA
AA
•@
@
@@
 
 
OnA
AA
• 
 
 
 
 
 
PODIn A
AA
•@
@
@
@
@
@
ODnA
AA•POnA
AA
•PODnA
AA
The first section of this note is dedicated to recalling known formulas to calculate the cardinal of the monoids
defined above. In the second section we consider the analogous question for the set of idempotents of those
monoids. Here we revise the known results and complete the study by computing the remaining cases.
1 Cardinals
Let PDn be the set of all order-reversing partial transformations of Xn and let IDn and Dn be the subsets of
PDn of all injective transformations and of all full transformations, respectively. Clearly, PODn = POn∪PDn
and so PODIn = POIn ∪ IDn and ODn = On ∪ Dn. Furthermore, POn ∩ PDn = {s ∈ PTn : | Im(s)| ≤ 1}
and so POIn ∩ IDn = {s ∈ In : | Im(s)| ≤ 1} and On ∩ Dn = {s ∈ Tn : | Im(s)| = 1}.
2
DM-FCT-UNL pre-prints
Consider the following order-reversing permutation of order two:
h =
(
1 2 · · · n− 1 n
n n− 1 · · · 2 1
)
.
To calculate the cardinals of PODn, PODIn and ODn, we will make use of the mapping ϕ : POn −→ PDn
defined by (s)ϕ = sh, for all s ∈ POn. Clearly, this mapping ϕ is a bijection and so we have |PDn| = |POn|.
On the other hand, ϕ maps On onto Dn and POIn onto IDn, whence |Dn| = |On| and |IDn| = |POIn|.
Now, we recall that Howie [8] showed that
|On| =
(
2n− 1
n− 1
)
and since we have |On ∩ Dn| = |{s ∈ Tn : | Im(s)| = 1}| = n, the next result followed.
Theorem 1.1 [5] |ODn| = 2
(
2n− 1
n− 1
)
− n. 
As |POIn ∩ IDn| = |{s ∈ In : | Im(s)| ≤ 1}| = n2 + 1 and Fernandes [2] proved that
|POIn| =
(
2n
n
)
,
we deduced the cardinal of PODIn.
Theorem 1.2 [4] |PODIn| = 2
(
2n
n
)
− n2 − 1. 
Next, we noticed that Gomes and Howie [7] established that
|POn| =
n∑
i=1
(
n
i
)(
n+ i− 1
i
)
+ 1
and using the fact that |POn ∩ PDn| = |{s ∈ PTn : | Im(s)| ≤ 1}| = n
∑n
i=1
(
n
i
)
+ 1, we computed |PODn|.
Theorem 1.3 [5] |PODn| =
n∑
i=1
(
n
i
)(
2
(
n+ i− 1
i
)
− n
)
+ 1. 
The cardinal of POPn was also calculated by the authors.
Theorem 1.4 [6] |POPn| = 1 + (2n − 1)n+
n∑
k=2
k
(
n
k
)2
2n−k. 
Denote by Porn the set of all orientation-reversing partial transformations of Xn. By definition, we have
PORn = POPn ∪ Porn. Catarino and Higgins proved the following fact:
Lemma 1.5 [1] Let a be a cyclic (resp., anti-cyclic) sequence. Then a is also anti-cyclic (resp., cyclic) if and
only if a has no more than two distinct values. 
This allows us to conclude that POPn ∩ Porn = {s ∈ POPn : | Im(s)| ≤ 2}. As the mapping Ψ :
POPn −→ Porn defined by (s)Ψ = sh, for all s ∈ POPn, is a bijection, we get |POPn| = |Porn| and so
|PORn| = 2|POPn| − |{s ∈ POPn | | Im(s)| ≤ 2}|. Therefore we obtain the cardinal of PORn.
Theorem 1.6 [6] |PORn| = 1 + (2n − 1)n+ 2
(
n
2
)2
2n−2 +
n∑
k=3
2k
(
n
k
)2
2n−k. 
3
DM-FCT-UNL pre-prints
The cardinal of POPIn, computed by Fernandes [3], is given by the following formula
|POPIn| = 1 + n2
(
2n
n
)
.
As, in this case Ψ maps POPIn onto the set of all injective orientation-reversing transformations, we conclude
that |PORIn| = 2|POPIn| − |{s ∈ POPIn | | Im(s)| ≤ 2}|. We then proved the following.
Theorem 1.7 [4] |PORIn| = 1 + n
(
2n
n
)
− n
2
2
(n2 − 2n+ 3). 
The cardinals of OPn and ORn were computed by McAlister [10] and, independently, by Catarino and
Higgins [1], who proved that
|OPn| = n
(
2n− 1
n− 1
)
− n(n− 1) and |ORn| = n
(
2n
n
)
− n
2
2
(n2 − 2n+ 5) + n.
Just to complete the picture recall that
|Cn| = n, |Sn| = n!, |In| =
n∑
k=0
(
n
k
)2
k!, |Tn| = nn and |PTn| = (n+ 1)n.
2 Number of idempotents
For a given monoid M , we denote by E(M) its set of idempotents.
First we will consider the order case. Let M ∈ {ODn,PODIn,PODn}. Let e ∈ E(M). As e2 = e and the
product of two order-preserving transformations or of two order-reversing transformations is an order-preserving
transformation, we conclude that e must be order-preserving. Thus E(ODn) = E(On) and E(PODn) =
E(POn).
In [8] Howie showed that
|E(On)| = F2n,
where Fn is the nth Fibonacci number.
Recall that, the Fibonacci numbers are recursively defined by
F0 = 0, F1 = 1, Fk+1 = Fk + Fk−1, k ≥ 1.
Another interesting sequence of numbers is the Lucas numbers, which are also recursively defined as follows
L0 = 2, L1 = 1, Lk+1 = Lk + Lk−1, k ≥ 1.
Fibonacci and Lucas numbers are intrinsically related, in fact, for any n ∈ N0,
Fn =
τn − θn
τ − θ and Ln = τ
n + θn,
where τ is the golden number and θ is its rational conjugate, that is τ = 1+
√
5
2 and θ =
1−√5
2 , and we have
F2n = FnLn. For further details, see e.g. [11].
In view of the above observations, we conclude that
|E(ODn)| = |E(On)| = F2n = τ
2n − θ2n
τ − θ .
Concerning the correspondent classes of partial transformations, we prove the following.
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DM-FCT-UNL pre-prints
Theorem 2.1 |E(PODn)| = |E(POn)| =
n∑
j=1
(
n
j
)
F2j + 1 =
n∑
j=1
(
n
j
)(
τ2j − θ2j
τ − θ
)
+ 1.
Proof. First, notice that an element s ∈ PTn is idempotent if and only if
Im(s) ⊆ Fix(s) = {x ∈ Dom(s) | xs = x}.
Next, observe that for each nonempty subset A of Xn, the number of idempotents of POn with domain A
coincides with |E(O|A|)|. Putting these facts together, we get
|E(PODn)| = |E(POn)| =
∑n
j=1
(
n
j
)|E(Oj)|+ 1 =∑nj=1 (nj)F2j + 1 =∑nj=1 (nj) ( τ2j−θ2jτ−θ )+ 1,
as required. 
We now look into the orientation case. Let M ∈ {ORn,PORIn,PORn}. Let e ∈ E(M). As for the order
case, the product of two orientation-preserving or of two orientation-reversing elements of M is an orientation-
preserving transformation, whence e preserves the orientation. Thus E(ORn) = E(OPn) and E(PORn) =
E(POPn).
Recall that Catarino and Higgins [1] showed that
|E(ORn)| = |E(OPn)| = L2n − (n2 − n+ 2) = τ2n + θ2n − (n2 − n+ 2).
We finish this note by computing the number of idempotents of POPn and of PORn.
Theorem 2.2 |E(PORn)|= |E(POPn)|=
n∑
j=1
(
n
j
)
[L2j − (j2− j+2)]+ 1=
n∑
j=1
(
n
j
)
[τ2j + θ2j − (j2− j+2)]+ 1.
Proof. Once again, as an element s ∈ PTn is idempotent if and only if Im(s) ⊆ Fix(s), and for each nonempty
subset A of Xn, the number of idempotents of POPn with domain A coincides with |E(OP |A|)|, we obtain
|E(PORn)| = |E(POPn)| =
∑n
j=1
(
n
j
)|E(OPj)|+ 1
=
∑n
j=1
(
n
j
)
[L2j − (j2 − j + 2)] + 1
=
∑n
j=1
(
n
j
) [
τ2j + θ2j − (j2 − j + 2)]+ 1,
as required. 
To complete the picture recall that |E(Cn)| = |E(Sn)| = 1, |E(POIn)| = |E(PODIn)| = |E(PORIn)| =
|E(In)| = 2n, |E(Tn)| =
∑n
j=1
(
n
j
)
jn−j and |E(PTn)| =
∑n
j=1
(
n
j
) (∑j
k=1
(
j
k
)
kj−k
)
+ 1.
References
[1] Paula M. Catarino and Peter M. Higgins, The monoid of orientation-preserving mappings on a chain,
Semigroup Forum 58 (1999), 190-206.
[2] Vı´tor H. Fernandes, Semigroups of order-preserving mappings on a finite chain: a new class of divisors,
Semigroup Forum 54 (1997), 230-236.
[3] Vı´tor H. Fernandes, The monoid of all injective orientation-preserving partial transformations on a finite
chain, Comm. Algebra 28 (2000), 3401-3426.
5
DM-FCT-UNL pre-prints
[4] Vı´tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Presentations for some monoids of
injective partial transformations on a finite chain, Southeast Asian Bull. Math. 28 (2004), 903-918.
[5] Vı´tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Congruences on monoids of order-
preserving or order-reversing transformations on a finite chain, Glasgow Math. J. 47 (2005), 413-424.
[6] Vı´tor H. Fernandes, Gracinda M. S. Gomes and Manuel M. Jesus, Congruences on monoids of transforma-
tions preserving the orientation on a finite chain, submitted.
[7] Gracinda M. S. Gomes and John M. Howie, On the ranks of certain semigroups of order-preserving trans-
formations, Semigroup Forum 45 (1992), 272-282.
[8] John M. Howie, Product of idempotents in certain semigroups of transformations, Proc. Edinburgh Math.
Soc. 17 (1971) 223–236.
[9] John M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, 1995.
[10] Donald McAlister, Semigroups generated by a group and an idempotent, Comm. Algebra 26 (1998), 515-
547.
[11] Fred S. Roberts, Applied Combinatorics, Prentice-Hall, Inc., Englewood Cliffs, 1984.
First and Third authors’ second address: Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universidade
Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal.
Second author’s second address: Departamento de Matema´tica, Faculdade de Cieˆncias, Universidade de Lisboa, 1749-016
Lisboa, Portugal.
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The cardinal and the idempotent number of various monoids of transformations on a finite chain

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