Infinite partition monoids
Let and be the partition monoid and symmetric group on an infinite set . We show that may be generated by together with two (but no fewer) additional partitions, and we classify the pairs for which is generated by . We also show that may be generated by the set of all idempotent partitions together with two (but no fewer) additional partitions. In fact, is generated by if and only if it is generated by . We also classify the pairs for which is generated by . Among other results, we show that any countable subset of is contained in a -generated subsemigroup of , and that the length function on is bounded with respect to any generating set.
Keywords: Partition monoids, Symmetric groups, Generators, Idempotents, Semigroup Bergman property, Sierpiński rank.
MSC: 20M20; 20M17.
Diagram algebras have been the focus of intense study since the introduction of the Brauer algebras  in 1937 and, subsequently, the Temperley-Lieb algebras  and Jones algebras . The partition algebras, originally introduced in the context of statistical mechanics , contain all of the above diagram algebras and so provide a unified framework in which to study diagram algebras more generally. Partition algebras may be thought of as twisted semigroup algebras of partition monoids, and many properties of the partition algebras may be deduced from corresponding properties of the associated monoids [8, 9, 19, 39]. Recent studies have also recognised partition monoids and some of their submonoids as key objects in the pseudovarieties of finite aperiodic monoids and semigroups with involution [2, 3, 4].
Partition monoids were originally defined as finite structures, but the definitions work equally well in the infinite case. Although most of the study of partition monoids so far has focused on the finite case, there have been a number of recent works on infinite partition monoids; for example, Green’s relations were characterized in , and the idempotent generated subsemigroups were described in . The purpose of this article is to continue the study of infinite partition monoids, and we investigate a number of problems inspired by analogous considerations in infinite transformation semigroup theory.
As noted in [8, 11], the partition monoids contain a number of important transformation semigroups as submonoids, including the symmetric groups, the full transformation semigroups, and the symmetric and dual symmetric inverse monoids; see [14, 17, 24, 27, 31, 32, 33] for background on these subsemigroups. Many studies of infinite transformation semigroups have concentrated on features concerning generation. It seems that the earliest result in this direction goes back to 1935, when Sierpiński  showed that for any infinite set and for any countable collection of functions , it is possible to find functions for which each of can be obtained by composing and in some order a certain number of times. In modern language, this result says that any countable subset of the full transformation semigroup is contained in a two-generator subsemigroup, or that the Sirepiński rank of infinite is equal to . (The Sierpiński rank of a semigroup is the minimal value of such that any countable subset of is contained in an -generator subsemigroup of , if such an exists, or otherwise.) Similar results exist for various other transformation semigroups [10, 20, 28, 36]; see also  for a recent survey.
The notion of Sierpiński rank is intimately connected to the idea of relative rank. The relative rank of a semigroup modulo a subset is defined to be the least cardinality of a subset of for which is equal to , the semigroup generated by . In the seminal paper on this subject  (see also ), it was shown that an infinite full transformation semigroup has relative rank modulo either the symmetric group or the set of all idempotents in . In that paper, the pairs of transformations that, together with (in the case of being a regular cardinal—see  for the singular case) or (for any infinite set ), generate all of were characterized. Again, these results have led to similar studies of other transformation semigroups [1, 10, 20, 22, 23].
Another closely related concept is the so-called semigroup Bergman property; a semigroup has this property if the length function for the semigroup is bounded with respect to any generating set (the bound may be different for different generating sets). The property is so named because of the seminal paper of Bergman , in which it was shown that the infinite symmetric groups have this property; in fact, Bergman showed that infinite symmetric groups have the corresponding property with respect to group generating sets, and the semigroup analogue was proved in . Further studies have investigated the semigroup Bergman property in the context of other transformation semigroups [10, 34, 36].
The goal of the present article is to investigate problems such as those above in the context of infinite partition monoids. The article is organised as follows. In Section 2, we define the partition monoids and outline some of their basic properties. In Section 3, we show that has relative rank modulo the symmetric group (Theorem 12) and then, in Section 4, we characterize the pairs for which is generated by . This characterization depends crucially on the nature of the cardinal ; we have three separate characterizations, according to whether is countable (Theorem 22), or regular but uncountable (Theorem 19), or singular (Theorem 25). In Section 5, we show that the relative rank of modulo the set of all idempotent partitions is also equal to (Theorem 30); in fact, the relative rank of modulo is equal to as well. Then, in Section 6, we show that for any , is generated by if and only if it is generated by , and we characterize all such pairs (Theorem 36). The characterization in this case does not depend on the cardinality of , but relies crucially on results of  describing the semigroups and . Finally, in Section 7, we apply the above results to show that has Sierpiński rank at most (Theorem 37), and also satisfies the semigroup Bergman property (Theorem 41).
All functions will be written to the right of their arguments, and functions will be composed from left to right. We write to indicate that is the disjoint union of and . We write for the set of natural numbers . Throughout, a statement such as “Let ” should be read as “Let and assume the map is a bijection”. We assume the Axiom of Choice throughout. If is an infinite set, we will say a family of subsets of is a moiety of if and for all . A cardinal is singular if there exists a set such that , where and for each , but ; otherwise, is regular. The only finite regular cardinals are , and . The smallest infinite singular cardinal is . See  for more details on singular and regular cardinals.
In this section, we recall the definition of the partition monoids , and revise some of their basic properties. We also introduce two submonoids, and , which will play a crucial role throughout our investigations, and we define a number of parameters associated to a partition that will allow for convenient statements of our results.
Let be a set, and a disjoint set in one-one correspondence with via a mapping . If we will write . A partition on is a collection of pairwise disjoint nonempty subsets of whose union is ; these subsets are called the blocks of the partition. The partition monoid on is the set of all partitions on , with a natural associative binary operation defined below. A block of a partition is said to be a transversal block if , or otherwise an upper (respectively, lower) nontransversal block if (respectively, ). If , we will write
to indicate that has transversal blocks (), upper nontransversal blocks (), and lower nontransversal blocks (). The indexing sets will sometimes be implied rather than explicit, for brevity; if they are distinct, they will generally be assumed to be disjoint. Sometimes we will use slight variants of this notation, but it should always be clear what is meant.
A partition may be represented as a graph on the vertex set ; edges are included so that the connected components of the graph correspond to the blocks of the partition. Of course such a graphical representation is not unique, but we regard two such graphs as equivalent if they have the same connected components. We will also generally identify a partition with any graph representing it. We think of the vertices from (respectively, ) as being the upper vertices (respectively, lower vertices), explaining our use of these words in relation to the nontransversal blocks. An example is given in Figure 1 for the partition , where . Although it is traditional to draw vertex directly above vertex , especially in the case of finite , this is not necessary; indeed, we will often be forced to abandon this tradition. It will also be convenient to sometimes identify a partition with its corresponding equivalence relation on , and write to indicate that belong to the same block of .
The rule for multiplication of partitions is best described in terms of the graphical representations. Let . Consider now a third set , disjoint from both and , and in bijection with both sets via the maps and . Let be the graph obtained from (a graph representing) simply by changing the label of each lower vertex to . Similarly, let be the graph obtained from by changing the label of each upper vertex to . Consider now the graph on the vertex set obtained by joining and together so that each lower vertex of is identified with the corresponding upper vertex of . Note that , which we call the product graph of and , may contain multiple edges. We define to be the partition that satisfies the property that belong to the same block of if and only if there is a path from to in . An example calculation (with finite) is given in Figure 2. (See also  for an equivalent formulation of the product; there was denoted , and called the composition semigroup on .)
This product is easily checked to be associative, and so gives the structure of a monoid; the identity element is the partition , which we denote by . A partition is a unit if and only if each block of is of the form for some . So it is clear that the group of units, which we denote by , is (isomorphic to) the symmetric group on . So, if and , we will write for “the image of under ”, by which we mean the unique element of such that is a block of .
A crucial aspect of the structure of is given by the map where is the result of “turning upside-down”. More precisely:
Note that if . The next lemma is proved easily, and collects the basic properties of the map that we will need. Essentially it states that is a regular -semigroup.
Let . Then
Among other things, these properties mean that the map is an anti-isomorphism of . This duality will allow us to shorten many proofs.
Next we record some notation and terminology. With this in mind, let . For , we denote the block of containing by . The domain and codomain of are defined to be the following subsets of :
|We also define the kernel and cokernel of to be the following equivalences on :|
Note that and .
Let . Then
, with equality if ,
, with equality if ,
, with equality if , and
, with equality if .
Clearly . Suppose . Let . If one of or belongs to , then so too does the other, and . So suppose . Then for some . Since , there exist such that , and , , and so on. But, since , it follows that . This then implies that , and . This completes the proof of (2.3).
We now define two submonoids of that will play a crucial role in what follows. Denote by the trivial equivalence (that is, the equality relation). Let
Note that and , and that .
The sets and are submonoids of . Further, is a right ideal of , and is a left ideal.
Next, let and . If , then , so that , and . If , then we similarly obtain .
A typical element of has the form
In what follows, we will shorten this to , or just . Accordingly, we will write for the partition
from . Note that if and , then , where and for each and ; see Figure 3. A similar rule holds for multiplication in .
We now define a number of parameters associated with a partition. With this in mind, let and write
For any cardinal , we define
Note that and . We also have identities such as if . It will also be convenient to write
The above parameters are natural extensions of those introduced in the context of transformation semigroups in  (see also [13, 25]). These parameters should not be confused with those introduced in , such as , , etc.
Let . Then
Note that each is an upper nontransversal block of , so . Suppose now that is an upper nontransversal block of but that for any . Then for some subset . Now, must have trivial intersection with each of the , or else would be contained in a transversal block of . But this implies that intersects at least one of the . In particular, there are at most such upper nontransversal blocks . Thus, .
The above-mentioned rule for multiplication in shows that if . A dual identity holds in .
For the following lemmas, recall that we count and as regular cardinals.
Let and and let be any cardinal. Then
if is regular,
if is regular.
Proof We just prove (7.1) and (7.2), since the others will follow by duality. Let and . Then , where and for each and . Clearly, for all , so , establishing (7.1). Next, suppose is regular. If for some , then either (i) , or (ii) for some . There are values of that satisfy (i), and at most values of that satisfy (ii). Thus, , establishing (7.2).
Let and and let be any cardinal. Then
if is regular,
if is regular.
Proof Again, it suffices to prove (8.1) and (8.2). Write and . Then , where and for each and . There are values of for which . It follows that . Next, suppose is regular, and that is such that . Then either (i) , or (ii) for some . There are values of for which (i) holds, and at most values of for which (ii) holds. Thus, .
The next lemma will be used on a number of occasions. There is a dual result, but we will not need to state it.
Let with , and let be any cardinal. Then there exists such that , , and for all .
Proof Let and put , noting that . We will consider two separate cases.
Case 1. First suppose . For each , choose some . Let be any permutation that extends the map , and put . Then, for each , is a lower nontransversal block of , and . Thus, .
Case 2. Now suppose . Let be a moiety of , and let be any permutation that extends any bijection . Then for any , is a lower nontransversal block of of size at least . It follows that .
In either case, , so that for all . And, in either case, is a consequence of (5.2).
3 Relative rank of modulo
Recall that the relative rank of a semigroup with respect to a subset , denoted , is the minimum cardinality of a subset such that . Our goal in this section is to show that ; see Theorem 12.
Recall that for and , we write for the block of containing . We will also write for the cardinality of . The next result shows that may be generated by along with just two additional partitions. See [25, Theorem 3.3] for the corresponding result for infinite transformation semigroups.
Let and be such that , and for all . Then .
Proof Consider an arbitrary partition
We will construct a permutation such that . The assumptions on allow us to write and , where for all . See Figure 4 for an illustration (the picture shows the basic “shape” of and , and is not meant to indicate that ). Let
So . Note that and, similarly, . We now proceed to construct in stages.
Stage 1. Fix . For each , let where and , and write . For each , let where and , and write . Now let be any bijection that extends the map It is easy to check that if is any permutation that extends , then is a block of . See Figure 5 for an illustration.