Results

We will now consider a number of submonoids in turn, giving the classification of their congru- ences. The total number of congruences of each monoid is shown in Table 5.36.

Monoid Size Number of congruences

Sn n! 3 In P n k=0 n k n! (n−k)! 3n−1 POIn 2nn n+ 1 Mn Pn_k=0 2₂n_kCk n+ 7 PPn C2n n+ 7 Bn (2n−1)!! 3₂n+5₂ or 3₂n+ 13 Jn Cn 1₂n+7₂ or 1₂n+ 7 PBn P n k=0 2n 2k (2k−1)!! 3n+ 7 Pn B2n 3n+ 7

Table 5.36: The number of congruences on various diagram monoids. Numbers shown are correct forn≥5.

We saw in Example 1.81 a way in which partial transformations lie in the partition monoid

Pn. We will therefore start with three monoids of partial transformations which embed into

Pn as submonoids: Sn, In, andPOIn. For all the monoids below, we assumen≥3, since any

lowernhas very few elements and is rather trivial to solve.

Symmetric group Sn

The symmetric group Sn is isomorphic to the subgroup ofPn consisting of all bipartitions of

its normal subgroups (see Section 3.1.2). These normal subgroups are well known: the trivial group {id}, the alternating group An, the whole symmetric group Sn itself, and uniquely in

the case thatn= 4, the Klein 4-group K4 =h(1 2)(3 4),(1 3)(2 4)i. For n≥3 these normal

subgroups (and hence these congruences) are all distinct.

Symmetric inverse monoid In

Recall that the symmetric inverse monoidIn consists of all the partial permutations of rank up

tonunder composition. This embeds intoPn as in Example 1.81 as the submonoid consisting

of all bipartitions with trivial kernel and cokernel.

The ideals ofIn form a chain with respect to containment, and are precisely the sets

Ik ={α∈ Mn: rankα≤k},

fork∈ {0, . . . , n}, as is the case forPn andMn.

The congruences of In were classified in [Lib53], and are reformulated in the context of

IN-pairs as follows.

Theorem 5.37 ([EMRT18, Theorem 4.1]). Let In be the inverse symmetric monoid of degree

n, for n≥0. The congruences of In form a chain, and are as follows:

• the Rees congruencesRk corresponding to the idealsIk, for k∈ {0, . . . , n};

• the congruences RI,N corresponding to the IN-pairs (Ik−1, N) for k ∈ {2, . . . , n} and

N ∈ {K4,Ak,Sk} being any non-trivial normal subgroup of Sk (the group isomorphic to

a maximal subgroup ofJk).

Note thatAk andSk will be used for everyk∈ {2, . . . , n}, butK4 will only be used when

k = 4. Since A2 is trivial, we have RI1,A2 = R1. Note also that, since there is only one

bipartition in In of rank 0, we have R0 = ∆In. As an example, the congruences ofI6 are as follows: ∆I6=R0⊂R1 ⊂RI1,S2 ⊂R2 ⊂RI2,A3⊂RI2,S3 ⊂R3 ⊂RI3,K4 ⊂RI3,A4 ⊂RI3,S4 ⊂R4 ⊂RI4,A5⊂RI4,S5 ⊂R5 ⊂RI5,A6⊂RI5,S6 ⊂R6=∇I6.

Order-preserving partial permutation monoidPOIn

The monoidPOInof all order-preserving partial permutations embeds intoPnas the submonoid

consisting of all the planar bipartitions with trivial kernel and cokernel. An element ofPOIn is

defined by its domain and codomain, soPOIncontains n_r

2 elements of rankr, orPn r=0 n r 2 = 2n n

Its ideals have the same description as forIn, and its congruences are all Rees. Hence the

congruences ofPOIn form the chain

∆POIn=R0⊂R1⊂R2⊂. . .⊂Rn=∇POIn. This is shown in [Fer01, Proposition 2.6].

Planar bipartition monoid PPn

The planar bipartition monoid PPn simply consists of all the planar bipartitions in Pn. It

therefore contains the Motzkin monoid, which has the additional restriction that a bipartition’s blocks have size 1 or 2. The planar bipartition monoidPPn has a number of elements equal to

the Catalan numberC2n [OEIS, A000108]. Its congruence lattice is in fact isomorphic to that

ofMn, and its congruences have the same descriptions (detailed in Theorem 5.23) [EMRT18,

§7].

Brauer monoid Bn

The Brauer monoidBn consists of all the bipartitions in Pn whose blocks all have size 2. The

Brauer monoid contains

(2n−1)!! = (2n−1)·(2n−3)· · ·5·3·1

elements in total [OEIS, A001147]. Since each block in Bn must have size 2, the monoid only

contains elements with rank equal to n (mod 2) – in particular, Bn never contains both an

element of rank 0 and an element of rank 1. For this reason, in classifying the congruences of

Bn, we must consider two different cases: one in whichnis odd, and one in whichnis even.

The odd case is by far the simpler. All bipartitions in Bn are odd in this case, so the

Rees congruences are {R1, R3, . . . , Rn}. We also have lifted congruences{δ1, λ1, ρ1} which are

defined in the same way as for Mn. Finally, via IN-pairs, we have RIk−2,Ak and RIk−2,Sk for eachk∈ {3,5, . . . , n}.

In the even case, there are many more congruences. All bipartitions are even, so the Rees congruences are{R0, R2, . . . , Rn}. We also have lifted congruences

{ζI0,∆, ζI0,LI0, ζI0,RI0, ζI2,∆, ζI2,LI0, ζI2,RI0},

which we denote, in a way similar to the Motzkin monoid, as{δ0, λ0, ρ0, δ2, λ2, ρ2}. Some more

congruences arise from IN-pairs: (I0,S2) gives rise to

δS2 =ζI0,S2,∆, λS2 =ζI0,S2,LI0, ρS2 =ζI0,S2,RI0, RI0,S2;

and (I2, K4) gives rise to

δK4 =ζI2,K4,∆, λK4 =ζI2,K4,LI0, ρK4 =ζI2,K4,RI0, RI2,K4.

Finally, we haveRIk−2,Ak andRIk−2,Sk for eachk∈ {4,6, . . . , n}.

Rn RI4,A6 R4 RI2,S4 RI2,A4 RI2,K4 ρK4 λK4 δK4 R2 ρ2 λ2 δ2 RI0,S2 ρS2 λS2 δS2 R0 ρ0 λ0 δ0 =∇Bn = ∆Bn Rn RI3,A5 R3 RI1,S3 RI1,A3 R1 λ1 ρ1 δ1 =∇Bn = ∆Bn

Figure 5.38: Congruence lattice of Bn forn ≥5 when n is odd (upper left) and even (lower

Jones monoid Jn

The Jones monoid Jn is the submonoid ofPn consisting of all planar bipartitions with blocks

of size 2. By this definition, we can see thatJn=PPn∩ Bn. Its size is given by the Catalan

number Cn [OEIS, A000108].

As with the Brauer monoid, we consider two different cases based on whethern is odd or even; however, the congruence lattices are much simpler. Ifnis odd, then the only congruences areδ1,λ1,ρ1, and the Rees congruences{R1, R3, . . . , Rn}. If, on the other hand,nis even, then

the congruence lattice is isomorphic to that of Mn/2, and its description can be obtained by

doubling each number in the description of that lattice. That is, the congruences are precisely

{δ0, δ2, λ0, λ2, ρ0, ρ2, R0, R2, . . . , Rn}.

These results are proven in [EMRT18, §9], and the lattices are shown in Figure 5.39. Rn R4 R2 λ2 R0 ρ2 λ0 δ2 ρ0 δ0 =∇Jn = ∆Jn Rn R5 R3 R1 λ1 ρ1 δ1 =∇Jn = ∆Jn

Figure 5.39: Congruence lattice ofJn forn≥4 when nis odd (left) and even (right).

Bipartition monoid Pn and partial Brauer monoid PBn

Finally, we can state the congruence lattice of the entire bipartition monoidPn. As in the case

of the Motzkin monoid, we have Rees congruences {R0, R1, . . . , Rn} and lifted congruences

{δ0, δ1, λ0, λ1, ρ0, ρ1}. The additional congruences onPn come from IN-pairs: the retractable

IN-pair (I1,S2) gives rise to congruences

and the non-retractable IN-pairs (Ik−1,Ak) and (Ik−1,Sk) give us the congruences RIk−1,Ak andRIk−1,Ak, fork∈ {3, . . . , n}. Uniquely fork= 4 we also have the IN-pair (I3, K4), yielding the congruence RI3,K4. These are all the congruences on Pn, as is proven in [EMRT18, §5].

The lattice is shown in Figure 5.40.

We should also mention the partial Brauer monoidPBn, the submonoid ofPn consisting of

all the bipartitions with blocks of size 1 or 2. It has

n X k=0 _2n 2k (2k−1)!!

elements [OEIS, A066223], as shown in [Hd14, 2.1]. Again we can see that this monoid contains the Motzkin monoid; in fact, it is clear from the definitions that Mn = PPn ∩ PBn. Its

congruence lattice has the same description as that of Pn [EMRT18,§6], and is therefore also

Rn R3 RI2,S3 RI2,A3 R2 RI1,S2 ρS2 λS2 δS2 R1 ρ1 λ1 δ1 R0 ρ0 λ0 δ0 =∇Pn = ∆Pn

Chapter 6

Principal factors and counting

congruences

In Chapter 5 we classified the congruences of the Motzkin monoid and several related diagram monoids. This classification was achieved by first calculating the congruence lattices for small values ofnusing the computational techniques described in Chapters 2 and 4, and then building up theory in order to prove a classification for generaln. In this chapter we present some more results about congruences that were obtained in a similar way, by first looking for patterns in computational results, and then extending the results and attempting to prove them for larger semigroups. Thelibsemigroupslibrary and theSemigroupsandsmallsemipackages forGAPwere used to carry out the initial computations [MT+_{18, M}+_{19, DM17, GAP18].}

6.1

Congruences of principal factors

In this section, we will consider an interesting decomposition of a semigroup related to itsJ- classes: a semigroup’s principal factors. After defining this construction, we will consider the principal factors of the full transformation monoid Tn, and classify their congruences. After

this, we will look at the principal factors of some other, somewhat similar monoids, and classify their congruences using similar principles.