5.4 Other monoids
6.1.3 Other semigroups
Now that we have considered the principal factors of the full transformation monoid, we can go on to consider the principal factors of some other semigroups related toTn, and classify their
congruences. The proofs are broadly similar to those for Tn, so we will only summarise the
arguments, highlighting the parts where they differ from those in Section 6.1.2. We start by extending our consideration of transformations to partial transformations and then to partial permuations; then we consider the three corresponding order-preserving submonoids.
Partial transformation monoid PTn
Recall that PTn is the monoid of all partial transformations on the setn={1, . . . , n}, that is,
all transformations on some subset ofn. In many respects a partial transformation behaves like a transformation: it has an image, a rank (the size of the image), and a kernel. However, we should also consider a partial transformation’s domain: the set of points which it maps. The
n= 1 n= 2 n= 3 n= 4 n= 5 k= 1 1 1 to 2 1 to 3 1 to 4 1 to 5 k= 2 – 3, 2, 1 19, 10, 1 85, 43, 1 301, 151, 1 k= 3 – – 7, 3, 2, 1 145, 49, 25, 1 1501, 501, 251, 1 k= 4 – – – 25, 7, 3, 2, 1 1201, 301, 101, 51, 1 k= 5 – – – – 121, 3, 2, 1 n= 6 n= 7 k= 1 1 to 6 1 to 7 k= 2 931, 466, 1 2647, 1324, 1 k= 3 10801, 3601, 1801, 1 63211, 21071, 10536, 1 k= 4 23401, 5851, 1951, 976, 1 294001, 73501, 24501, 12251, 1 k= 5 10801, 181, 91, 1 352801, 5881, 2941, 1 k= 6 721, 3, 2, 1 105841, 295, 148, 1 k= 7 – 5041, 3, 2, 1
Table 6.6: Number of classes of the congruences on the principal factors ofTn, fornup to 7.
kernel is a partition of the domain. Using these definitions, the Green’s relations of PTn are
described in the same way as those ofTn: theD-classes are determined by rank, theL-classes
by image, theR-classes by kernel, and theH-classes by image and kernel. Idempotents are also described in the same way: a partial transformationα∈ PTn is an idempotent if and only
if its image is a cross-section of its kernel.
The D-classes of PTn are somewhat different from Tn. Firstly, there exists an element
0 =− − · · · −1 2 · · ·n with rank 0, so we have an additionalD-class Dn
0. For a given rankk, there
are still nk
possible images with sizek, so theD-classDn k has
n k
L
-classes, likeTn. However,
the possibility of points not being in the domain means that there are not justS(n, k) possible kernels, but S(n+ 1, k+ 1); the intuition behind this is that, instead of considering all k- partitions of{1, . . . , n}, we are considering allk+ 1-partitions of{1, . . . , n+ 1}, where the class containingn+ 1 represents those points outside the domain.
Lemma 6.3 holds true forPTn – in fact, both parts hold even whenk= 0 or 1. Fork= 0
we simply observe thatD0n is trivial, and the two statements follow. The proof given for (i) is
sufficient for PTn when k ≥1, so (i) is proven. For (ii), the proof given is only sufficient for
PTn whenk≥2. For k= 1, it is proven as follows. We must have I1={i}andI2={j} with
i6=j; simply letP =
{i} , a cross-section ofI1 but notI2. This is not possible inTnbecause
any kernel has to contain all the elements ofnsomewhere; but by missing out points from the domain, it is possible inPTn.
Since Lemma 6.3 holds for allk, we haveεI = ∆I andεΛ= ∆Λin each principal factorDnk.
Hence every linked triple onDkn must have the form (N,∆,∆). Each groupH-class inDnk is isomorphic to Sk, since it corresponds to all the ways of mapping the k classes of the kernel
subgroups ofSk. The congruences are summarised in Table 6.7, wherehnk is equal to S(n+ 1, k+ 1) n k , the number ofH-classes inDn
k.
k Congruences of Dn
k Number Number of classes
0 [{id}] 1 1 1 [{id}],∇ 2 hn1 + 1,1 2 [{id}],[S2],∇ 3 2hn2 + 1, hn2 + 1,1 3 [{id}],[A3],[S3],∇ 4 6hn3+ 1,2hn3 + 1, hn3 + 1,1 4 [{id}],[K4],[A4],[S4],∇ 5 24hn4 + 1,6hn4+ 1,2hn4 + 1, hn4 + 1,1 ≥5 [{id}],[Ak],[Sk],∇ 4 k!hnk+ 1,2h n k + 1, h n k + 1,1
Table 6.7: Congruences of the principal factors ofPTn or In.
Symmetric inverse monoid In
Recall that the symmetric inverse monoidIn consists of all partial permutations on the setn;
that is,In is the submonoid ofPTn consisting of the injective maps. The Green’s relations of
In are determined by rank, image and kernel, as forPTn, but we can think of theR relation
in a slightly simpler way. Since each element of In is a partial permutation, its kernel must
be the diagonal relation on the domain; hence, two elements are R-related if and only if they have the same domain. This creates a certain symmetry between theL andRrelations: if an element is written in two-row notation, the set of points in the top row determine itsR-class, and the set of points in the bottom row determine itsL-class.
This symmetry makes the classification of the principal factors’ congruences quite straight- forward. The D-class Dn
k of elements with rank k contains n k
L
-classes and nkR
-classes. The idempotents ofIn are simply the identity maps (that is, the elements αsuch that iα=i
for all i∈domα) so anH-class is a group if and only if its image and its domain are equal. Hence eachL-class and eachR-class contains precisely one groupH-class. This is enough to prove the whole of Lemma 6.3, for all k from 0 to n. Hence, as for PTn, all linked triples on
Dn
k are of the form (N,∆,∆).
There arek! elements with a given image and domain of sizek, and if the image and domain are equal they form a group isomorphic toSk, so like PTn we have that the choices forN are
all the normal subgroups ofSk. The result is that the congruences of the principal factors ofIn
have the same description as those of the principal factors ofPTn. They can be seen in Table
6.7, where the number ofH-classeshn
k is in this case given by n k
2 .
Order-preserving partial transformations POn
Recall that a partial transformationα∈ PTnis calledorder-preservingif, for pointsi, j∈domα,
we havei≤jif and only ifiα≤jα. The order-preserving partial transformations inPTnform
The Green’s relations have the same description as in PTn, being based on rank, image
and kernel. However, some partitions of ndo not occur as kernels inPOn, since they cannot
preserve order. LetP be a partition ofnwhich contains three pointsi < j < ksuch thatiand kare in the same kernel class, andj is in a different kernel class. Any partial transformationα with kernelP cannot preserve order, since it must have eitherjα < iα=kαoriα=kα < jα. A partition is a valid kernel forPOn if and only if it observes the following rule: a point i is
either in the same class as i−1, or it is the lowest point in its class. Hence, there are not S(n+ 1, k+ 1) R-classes in Dkn, as there are forPTn. The actual number ofR-classes in Dkn
is given by n X i=k n i i−1 k−1 ,
as shown in [LU04, Lemma 4.1]. This can be understood in the following way. Since an element in Dnk has rankk, the domain can have any size fromk ton. Given a domain sizei, there are
n i
choices for the domain. Once we have chosen a domain, we must split theidomain points into classes. By the above description of a valid kernel, this involves simply choosing which i of the n points are the lowest in their kernel-class. Point 1 must be lowest, so we have ni−−11 choices. The exception to this rule isDn
0, which simply has oneR-class.
Idempotents have the same characterisation as forPTn. Lemma 6.3 holds for allk from 0
to n, as follows. Dn
0 is trivial, so both statements hold for k= 0. The proof given for (i) is
sufficient in this case for allk≥1. To prove (ii) fork= 1, we use the same approach described for PTn. To prove (ii) for k≥2, let I1 and I2 be the images ofL1 and L2 respectively; if we
take the kernel ∆I1, then I1 is a cross-section of it butI2 is not, so theR-class corresponding
to that kernel satisfies the requirement.
Perhaps the most important difference betweenPOn andPTn is that inPOn a given kernel
and image determines a single element, not k! elements, since order must be preserved. This means that the underlying group ofDn
k is notSk, but simply the trivial group{id}. This result
puts the principal factors Dn
k into the category of congruence-free semigroups by Proposition
6.11, meaning that the only congruences onDn
k are ∆ and∇. Indeed, the only linked triple of
Dn
k is ({id},∆,∆), corresponding to the trivial congruence. This result is summarised in Table
6.8, where hnk is the number of H-classes inDnk, given by n k n X i=k n i i−1 k−1 . k Congruences of Dn
k Number Number of classes
0 [{id}] 1 1
≥1 [{id}],∇ 2 hnk+ 1,1 Table 6.8: Congruences of the principal factors ofPOn orPOIn.
Order-preserving partial permutations POIn
Next we consider the order-preserving partial permuations, which form the monoid POIn =
Lemma 6.3 is proven in the same way as for In, and applies to all k from 0 to n. Since the
kernel of a partial permutation is always a diagonal relation, we do not encounter any kernels which cannot preserve order; hence the Green’s class structure ofPOIn is isomorphic to that of
In. In particular, there are nk 2
H-classes inDn
k. The main difference betweenPOIn andIn
is that eachH-class contains just one element, since each domain–image pair defines only one order-preserving element. Hence the underlying group of Dn
k is the trivial group {id}, and so
POIn is congruence-free like POn. This information is summarised in Table 6.8, where in this
case the number ofH-classeshnk is equal to nk2.
Order-preserving transformationsOn
Finally, we consider the submonoid of Tn consisting of the order-preserving transformations,
On=Tn∩ POn. SinceOn consists of transformations, an element’s kernel includes every point
inn, as forTn. Its Green’s relationsL,RandDare again based on image, kernel and rank, as
forTn, so we do not have aD-classDn0. SomeR-classes inTnare not presentOn, since certain
kernels cannot preserve order: like inPOn, the valid kernels are those such that a pointieither
is in the same class as i−1, or is minimal in its class. Hence Dn
k contains n−1 k−1
R
-classes, since our only choice is which k−1 of the n−1 points innare minimal, apart from 1. There are still nkL
-classes, as inTn. Lemma 6.3 applies toOn in the same way as it applies toTn,
with statement (ii) only applying whenk >1. Hence the linked triples fork≥2 have the form (N,∆,∆) while the linked triples fork= 1 have the form (N,∆,T) for other possible values ofT.
Two elements are H-related if and only if they share the same image and kernel. Since all elements are order-preserving, there is only one choice of element for a given image and kernel; hence On is H-trivial. So the only choice of N for linked triples is the trivial group
{id}. Hence, whenk ≥ 2 the only linked triple on Dn
k is ({id},∆,∆), corresponding to the
trivial congruence; if k = 1, as forTn, we have a linked triple ({id},∆,T) for any relation T
on thenL-classes ofDn
1. The congruences are summarised in Table 6.9, where the number of
H-classeshn k is given by n k n−1 k−1 . k Congruences of Dn
k Number Number of classes
1 [{id},∆I,T](∀T) Bn from 1 ton
≥2 [{id}],∇ 2 hnk+ 1,1 Table 6.9: Congruences of the principal factors ofOn.
6.1.4
Further work
We have considered the monoids of partial transformationsPTn, transformationsTn, and partial
permutations In, and for each of those monoids we have considered the submonoid of order-
preserving elements. In the future, the ideas presented here could perhaps be extended to other similar submonoids of these three, such as the following submonoids, considered in [EKMW18,
§1.2]:
• the monoids of order-preserving or order-reversing elements: PODn, ODn, and PODIn
• the monoids oforientation-preserving elements: POPn,OPn, andPOPIn respectively;
• the monoids of orientation-preserving or or orientation-reversing elements: PORn, ORn,
andPORIn respectively.
We could also consider some important monoids which do not consist of partial transforma- tions. After the results in Chapter 5, it would be interesting to learn about the congruences of the principal factors of the Motzkin monoid and other bipartition monoids such asPn. These
monoids are not as straightforward as the ones we have so far considered; certainly, identifying the idempotents in a semigroup of bipartitions is more complicated than for partial transfor- mations [DEE+15, Theorem 5]. However, it is possible that their principal factors’ congruences
could be classified in a similar way.