2.1 Partitions

Let $\Omega $ be a finite set. The set of all partitions of $\Omega $ is partially ordered by refinement: $\Pi _1\preccurlyeq \Pi _2$ if each part of $\Pi _1$ is contained in a part of $\Pi _2$ . With this order, the partitions form a lattice (a partially ordered set in which any two elements have a greatest lower bound or meet, and a least upper bound or join): the meet (also called infimum) $\Pi _1\wedge \Pi _2$ is the partition whose parts are all nonempty intersections of parts of $\Pi _1$ and $\Pi _2$ , and the join (also called supremum) $\Pi _1\vee \Pi _2$ is the partition in which the part containing $\alpha $ consists of all points of $\Omega $ that can be reached from $\alpha $ by moving alternately within a part of $\Pi _1$ and within a part of $\Pi _2$ .

Partitions can be considered also as equivalence relations. The composition $R_1\circ R_2$ of two relations $R_1$ and $R_2$ is the relation in which $\alpha $ and $\beta $ are related if and only if there exists $\gamma $ with $(\alpha ,\gamma )\in R_1$ and $(\gamma ,\beta )\in R_2$ .

In view of the natural correspondence between partitions and equivalence relations, we abuse notation by talking about the join $R_1\vee R_2$ of two equivalence relations, or the composition $\Pi _1\circ \Pi _2$ of two partitions.

This result was first proved in [Reference Dubreil and Dubreil-Jacotin18].

2.2 Lattices

A finite lattice is conveniently represented by its Hasse diagram: this is the plane diagram with a dot for each lattice element; if $a\prec b$ then b is higher than a in the plane; and if b covers a (that is, $a\prec b$ but there is no element c with $a\prec c\prec b$ ), then an edge joins a to b.

In a lattice, the modular law states that

$$\begin{align*}a\preccurlyeq c \text{ implies }a\vee(b\wedge c)=(a\vee b)\wedge c. \end{align*}$$

A lattice L is modular if this holds for all $a,b,c\in L$ .

Proof. We are required to prove that $\Phi \preccurlyeq \Psi $ implies $\Phi \vee (\Xi \wedge \Psi )=(\Phi \vee \Xi )\wedge \Psi $ . In Figure 1, the dots represent points in $\Omega $ . Each edge is labelled by a partition of $\Omega $ . If an edge labelled $\Phi $ joins points $\alpha $ and $\beta $ , this means that $\alpha $ and $\beta $ are in the same part of $\Phi $ ; and similarly for $\Xi $ and $\Psi $ .

Figure 1 The modular law for commuting partitions.

Since $\Phi \preccurlyeq \Psi $ , any $\Phi $ - $\Psi $ path can be replaced by a single $\Psi $ edge. So, considering the paths from $\alpha $ to $\gamma $ in the diagram on the left shows that $\Phi \vee (\Xi \wedge \Psi )\preccurlyeq (\Phi \vee \Xi )\wedge \Psi $ . Also, on the right, the $\Psi $ - $\Phi $ path from $\theta $ to $\eta $ implies that there is a $\Psi $ edge between them. Thus there is a $\Xi \wedge \Psi $ path from $\theta $ to $\eta $ , and hence a $(\Xi \wedge \Psi )\vee \Phi $ path from $\theta $ to $\zeta $ : this gives the reverse inequality.

Proposition 2.2 is Theorem 9.11 in [Reference Bailey5] and Proposition 8 in [Reference Finberg, Mainetti and Rota19].

A lattice is distributive if it satisfies the conditions

$$ \begin{align*} (a\vee b)\wedge c &= (a\wedge c)\vee(b\wedge c),\\ (a\wedge b)\vee c &= (a\vee c)\wedge(b\vee c), \end{align*} $$

for all $a,b,c$ .

Proof. (a) Suppose that the first law above holds. Then

$$ \begin{align*} (a\vee c)\wedge(b\vee c) &= ((a\vee c)\wedge b)\vee((a\vee c)\wedge c)\\ &=(a\wedge b)\vee(c\wedge b)\vee c\\ &=(a\wedge b)\vee c. \end{align*}$$

The proof of the other implication is similar.

(b) Suppose that L is distributive and let $a,b,c\in L$ with $a\preccurlyeq c$ . Then

$$\begin{align*}a\vee(b\wedge c) = (a\vee b)\wedge(a\vee c)=(a\vee b)\wedge c,\end{align*}$$

since $a\preccurlyeq c$ implies $a\vee c=c$ .

Proposition 2.3 is a standard result in lattice theory and appears in [Reference Davey and Priestley16] as Lemmas 4.2 and 4.3.

The fundamental theorem on distributive lattices states that every finite distributive lattice is isomorphic to a sublattice of the Boolean lattice of all subsets of a finite set. More precisely, a down-set in a partially ordered set $(M,{\sqsubseteq })$ is a subset D of M with the property that, if $m\in D$ and $m'\sqsubseteq m$ , then $m'\in D$ . The down-sets form a lattice under the operations of intersection and union.

A proof of this theorem is in [Reference Cameron12, p. 192]. We sometimes abbreviate ‘join-indecomposable’ to JI.

In particular, if M is an antichain (a poset in which any two elements are incomparable), then every subset is a down-set, and the corresponding lattice is the Boolean lattice on M.

There are well-known characterisations of these classes of lattices. The Hasse diagrams of $P_5$ and $N_3$ are shown in Figure 2.

Figure 2 The lattices $P_5$ (left) and $N_3$ (right).

The proof of this theorem can be found in [Reference Davey and Priestley16, p. 134].

2.3 Orthogonal block structures

The next definition comes from experimental design in statistics: see the discussion in Section 3. Our treatment follows [Reference Bailey5].

An orthogonal block structure $(\Omega ,\mathcal {B})$ consists of a collection $\mathcal {B}$ of partitions of a single set $\Omega $ satisfying the conditions

1. (a) $\mathcal {B}$ is a sublattice of the partition lattice (that is, closed under meet and join);

2. (b) $\mathcal {B}$ contains the two extreme partitions (the equality partition E whose parts are singletons, and the universal partition U with just one part);

3. (c) every partition in $\mathcal {B}$ is uniform (that is, has all parts of the same size);

4. (d) any two partitions in $\mathcal {B}$ commute.

The set $\mathcal {B} = \{E,U\}$ is an orthogonal block structure, which we call trivial.

We remark that the definition in [Reference Bailey5, Chapter 6] has a more complicated condition in place of our condition (d). With any partition $\Pi $ is associated a subspace $V_\Pi $ of the vector space $\mathbb {R}^\Omega $ consisting of functions which are constant on the parts of $\Pi $ , and the operator $P_\Pi $ of orthogonal projection of $\mathbb {R}^\Omega $ onto $V_\Pi $ ; two partitions $\Pi _1$ and $\Pi _2$ are said to be orthogonal if $P_{\Pi _1}$ and $P_{\Pi _2}$ commute. The remark at the top of page 153 of [Reference Bailey5] notes that, in the presence of conditions (a)–(c), this is equivalent to our simpler condition (d).

An association scheme on $\Omega $ is a partition of $\Omega ^2$ into symmetric relations $S_0,S_1,\ldots ,S_r$ having the properties that $S_0$ is the relation of equality and that the span over $\mathbb {R}$ of the zero-one relation matrices is an algebra. (Combinatorially this means that, given $i,j,k\in \{0,\ldots ,r\}$ and $\alpha $ , $\beta \in \Omega $ with $(\alpha ,\beta )\in S_k$ , the number $p_{ij}^k$ of elements $\gamma \in \Omega $ such that $(\alpha ,\gamma )\in S_i$ and $(\gamma ,\beta )\in S_j$ is independent of the choice of $(\alpha ,\beta )\in S_k$ , depending only on $i,j,k$ .)

An orthogonal block structure gives rise to an association scheme as follows. Let $R_0$ , $R_1$ , …, $R_t$ be equivalence relations forming an OBS. For each i, let

$$\begin{align*}S_i=R_i\setminus\bigcup_{j:R_j\subset R_i}R_j.\end{align*}$$

Then the nonempty relations $S_i$ are symmetric and partition $\Omega ^2$ ; after removing the empty ones and renumbering, we obtain an association scheme.

The nonequality relations in an association scheme are often thought of as graphs. We remark that, while in the association scheme associated with a primitive permutation group, all these graphs are connected, the association scheme associated with an orthogonal block structure is very different: all the graphs, except possibly the one associated with the universal relation U, are disconnected.

Note that, if two OBSs are isomorphic, then the association schemes obtained in this way are also isomorphic. The converse, however, is false, as the following example shows.

2.4 Crossing and nesting

Two methods of constructing new OBSs from old, both widely used in experimental design, are crossing and nesting, defined as follows.

Let $\mathcal {P}_1=(\Omega _1,\mathcal {B}_1)$ and $\mathcal {P}_2=(\Omega _2,\mathcal {B}_2)$ be orthogonal block structures. We think of the elements of $\mathcal {B}_1$ and $\mathcal {B}_2$ as equivalence relations. In each construction, we build a new OBS on $\Omega _1\times \Omega _2$ . For each pair $R_1\in \mathcal {B}_1$ and $R_2\in \mathcal {B}_2$ , we define a relation $R_1\times R_2$ to hold between two pairs $(\alpha _1,\alpha _2)$ and $(\beta _1,\beta _2)$ if and only if $(\alpha _1,\beta _1)\in R_1$ and $(\alpha _2,\beta _2)\in R_2$ . It is clear that $R_1\times R_2$ is an equivalence relation.

The first method uses the set of equivalence relations

$$\begin{align*}\{ R_1 \times R_2: R_1 \in \mathcal{B}_1,\ R_2\in\mathcal{B}_2\}.\end{align*}$$

This gives the set $\mathcal {B}_1 \times \mathcal {B}_2$ of equivalence relations on $\Omega _1 \times \Omega _2$ . This is called crossing $\mathcal {P}_1$ and $\mathcal {P}_2$ , and written $\mathcal {P}_1\times \mathcal {P}_2$ .

The second method uses the set of equivalence relations

$$\begin{align*}\{R_1 \times U_2: R_1\in \mathcal{B}_1\} \cup \{E_1\times R_2: R_2 \in \mathcal{B}_2\}, \end{align*}$$

where $U_2$ is the universal relation in $\Omega _2$ and $E_1$ is the equality relation in $\Omega _1$ . This is called nesting $\mathcal {P}_2$ within $\mathcal {P}_1$ , and written as $\mathcal {P}_1/\mathcal {P}_2$ .

Of course, the roles of $\mathcal {P}_1$ and $\mathcal {P}_2$ can be reversed, to give $\mathcal {P}_2/\mathcal {P}_1$ , with $\mathcal {P}_1$ nested within $\mathcal {P}_2$ .

It is straightforward to show that, if $\mathcal {P}_1$ and $\mathcal {P}_2$ are both closed under taking suprema and taking infima, then so are $\mathcal {P}_1 \times \mathcal {P}_2$ , $\mathcal {P}_1/\mathcal {P}_2$ and $\mathcal {P}_2/\mathcal {P}_1$ .

If $R_1$ and $R_3$ are in $\mathcal {B}_1$ and $R_2$ and $R_4$ are in $\mathcal {B}_2$ then $(R_1\circ R_3) \times (R_2\circ R_4)= (R_1\times R_2) \circ (R_3 \times R_4)$ . Therefore, since every two equivalence relations in $\mathcal {B}_1$ commute and every two equivalence relations in $\mathcal {B}_2$ commute, then the same is true for every two equivalence relations in each of $\mathcal {P}_1 \times \mathcal {P}_2$ , $\mathcal {P}_1/\mathcal {P}_2$ and $\mathcal {P}_2/\mathcal {P}_1$ .

Let $G\leq \mathrm {Sym}(\Gamma )$ and $H\leq \mathrm {Sym}(\Delta )$ be transitive permutation groups. Recall that the product action of the direct product $G\times H$ on $\Gamma \times \Delta $ is the action defined by $(\gamma , \delta )(g, h) = (\gamma g, \delta h)$ for all $(\gamma , \delta )\in \Gamma \times \Delta $ and $(g, h)\in G\times H$ . For permutation group theorists, note the similarities between crossing and nesting on one hand, and direct product (with product action) and wreath product (with imprimitive action) on the other. Statisticians call the results of crossing and nesting trivial OBSs row-column structures and block structures respectively.

Nelder [Reference Nelder29] introduced the class of orthogonal block structures which can be obtained from trivial structures by repeatedly crossing and nesting, and called them simple orthogonal block structures. See Section 3.

2.5 Poset block structures

There is a class of OBSs, more general than the simple ones, effectively introduced in [Reference Kempthorne, Zyskind, Addelman, Throckmorton and White26], and now called poset block structures, which we define.

A poset block structure is an orthogonal block structure in which the lattice of partitions is distributive. (We have seen in Proposition 2.3 that the distributive law is stronger than the modular law.)

Using the Fundamental Theorem on Distributive Lattices (Theorem 2.4), we can turn this abstract definition into something more useful. Recall that a distributive lattice L is the lattice of down-sets in a poset $(M,\sqsubseteq )$ , where M can be recovered from L as the set of nonzero join-indecomposable elements (that is, JI elements different from E). Put $N=|M|$ . Now we attach a finite set $\Omega _i$ of size $n_i>1$ to each element $m_i\in M$ , and take $\Omega $ to be the Cartesian product of the sets $\Omega _i$ for all $m_i\in M$ . Now we need to define a partition $\Pi _D$ for each down-set D in M. This is done as follows. Define a relation $R_D$ on $\Omega $ by

$$\begin{align*}R_D((\alpha_1,\ldots,\alpha_N),(\beta_1,\ldots,\beta_N))\Leftrightarrow(\forall m_i\notin D) (\alpha_i=\beta_i),\end{align*}$$

where $\alpha _i,\beta _i\in \Omega _i$ for all $m_i\in M$ . Then $R_D$ is an equivalence relation on $\Omega $ , and we let $\Pi _D$ be the corresponding partition. (The appearance of the poset M explains the name poset block structures.)

It is straightforward to check that

1. (a) the partitions E and U of $\Omega $ correspond to the empty set and the whole of M;

2. (b) if $D_1$ and $D_2$ are down-sets in M, then

   $$\begin{align*}\Pi_{D_1\cap D_2}=\Pi_{D_1}\wedge\Pi_{D_2}\text{ and } \Pi_{D_1\cup D_2}=\Pi_{D_1}\vee\Pi_{D_2}. \end{align*}$$

   So the partitions $\Pi _D$ form a lattice isomorphic to the given lattice L.

This is proved in [Reference Bailey4, Reference Speed, Bailey, Schultz, Praeger and Sullivan34], where it is shown that every poset block structure (according to our definition) is given by this construction.

At this point, we mention a paper by Yan [Reference Yan38], whose title suggests that it concerns distributive lattices of commuting equivalence relations. In fact, both her hypotheses and her conclusion are much stronger than ours. In the case of uniform partitions, her theorem asserts the following: if $\Pi _1$ and $\Pi _2$ are commuting uniform equivalence relations such that every equivalence relation $\Psi $ which commutes with both of them associates with them, in the sense that

$$\begin{align*}\Psi\wedge(\Pi_1\vee\Pi_2)=(\Psi\wedge\Pi_1)\vee(\Psi\wedge\Pi_2),\end{align*}$$

then $\Pi _1$ and $\Pi _2$ are comparable in the partial order. (This does not say that every distributive lattice of commuting partitions is a chain.)

2.6 Generalised wreath products

Closely related to poset block structures is the notion of generalised wreath product. We now define this, following the notation used in [Reference Bailey, Praeger, Rowley and Speed9].

We write $\Omega ^i$ for the Cartesian product $\prod _{j \in A(i)} \Omega _j$ and $\pi ^i$ for the natural projection from $\Omega $ onto $\prod _{j\in A(i)}\Omega _j$ . Finally, for every $m_i\in M$ , let $G(m_i)$ be a permutation group on $\Omega _i$ , and let $F_i$ denote the set of all functions from $\Omega ^i$ into $G(m_i)$ . Thus, if $f_i\in F_i$ , then $f_i$ allocates a permutation in $G(m_i)$ to each element of $\Omega ^i$ .

The generalised wreath product G of the groups $G(m_1), \ldots , G(m_N)$ over the poset M is the group $\prod _{i = 1}^N F_i$ , and it acts on $\Omega $ in the following way: if $\omega = (\omega _1,\ldots ,\omega _N) \in \Omega $ and $f = (f_1, \ldots , f_N) \in G$ , then

$$\begin{align*}(\omega f)_i = \omega_i(\omega\pi^i f_i) \end{align*}$$

for $i=1,\ldots ,N$ .

We note that, if M is the $2$ -element antichain $\{m_1,m_2\}$ , then the generalised wreath product of $G(m_1)$ and $G(m_2)$ is their direct product; while if M is a $2$ -element chain, with $m_1\sqsubset m_2$ , then G is the wreath product $G(m_1)\wr G(m_2)$ , in its imprimitive action.

The next result gives the automorphism group of a poset block structure.

This is proved in [Reference Bailey, Praeger, Rowley and Speed9].

The operations of crossing and nesting preserve the class of poset block structures: crossing corresponds to taking the disjoint union of the two posets (with no comparability between them); nesting corresponds to taking the ordered sum (with every element of the second poset below every element of the first).

Proposition 2.7 shows that poset block structures always have large automorphism groups. By contrast, orthogonal block structures may have no nontrivial automorphisms at all. Let L be a Latin square, with $\Omega $ the set of positions. Take the two trivial partitions and the three partitions into rows, columns and entries. Automorphisms of this structure are known as autotopisms in the Latin square literature; it is known that almost all Latin squares have trivial autotopism group: see [Reference Cameron14, Reference McKay and Wanless28].

3.1 Fisher and Yates at Rothamsted

Ronald Fisher was the first statistician at Rothamsted Experimental Station, working there from 1919 to 1933: see [Reference Bailey, Zack and Waszek6]. He advocated two, fairly simple, blocking structures. In the first, called a block design, the $bk$ plots were partitioned into b blocks of size k, thus giving the orthogonal block structure $\underline {\underline {b}}/\underline {\underline {k}}$ . In the second, called a Latin square, the $n^2$ plots formed a square array with n rows and n columns, to which n treatments were applied in such a way that each treatment occurred once in each row and once in each column. Ignoring the treatments, this gives the orthogonal block structure $\underline {\underline {n}}\times \underline {\underline {n}}$ .

Frank Yates worked in the Statistics Department at Rothamsted Experimental Station from 1931 until 1968: see [Reference Bailey, Zack and Waszek6]. He gradually developed more and more complicated block structures for designed experiments. His paper on ‘Complex Experiments’ [Reference Yates39], read to the Royal Statistical Society in 1935, covers many of these. After describing block designs and Latin squares, he proposes ‘splitting of plots’ (page 197) into subplots in both cases. If the number of subplots per plot is s, this leads to the orthogonal block structures $\underline {\underline {b}}/\underline {\underline {k}}/\underline {\underline {s}}$ and $(\underline {\underline {n}}\times \underline {\underline {n}})/\underline {\underline {s}}$ (treatments are ignored in these block structures). These are all based on partially ordered sets (although he did not use this terminology), as shown in Figure 3.

Figure 3 Orthogonal block structures mentioned by Yates in [Reference Yates39].

Yates also suggests ‘two $4\times 4$ Latin squares with subplots’ (page 201), which gives the orthogonal block structure $\underline {\underline {2}}/(\underline {\underline {4}} \times \underline {\underline {4}}) /\underline {\underline {2}}$ ; splitting each row of an $\underline {\underline {r}}\times \underline {\underline {c}}$ rectangle into two subrows, which gives the orthogonal block structure $(\underline {\underline {r}}/\underline {\underline {2}})\times \underline {\underline {c}}$ (page 202); and a collection of four $5 \times 5$ Latin squares (page 218), which gives the orthogonal block structure $\underline {\underline {4}}/(\underline {\underline {5}}\times \underline {\underline {5}})$ . These are shown in Figure 4.

Figure 4 More orthogonal block structures mentioned by Yates.

3.2 Nelder’s simple orthogonal block structures

John Nelder worked in the Statistics Section of the UK’s National Vegetable Research Station from 1951 to 1968. In two papers [Reference Nelder29, Reference Nelder30] in 1965 he introduced the class of orthogonal block structures which can be obtained from trivial structures by repeated crossing and nesting, and called them simple orthogonal block structures. In that year, he also visited CSIRO (the Commonwealth Scientific and Industrial Research Organisation) at the Waite Campus of the University of Adelaide in South Australia, where he worked with Graham Wilkinson to start developing the statistical software GenStat. He and colleagues developed GenStat further while he was Head of the Statistics Department at Rothamsted Experimental Station from 1968 to 1984. The benefit of iterated crossing and nesting is that each block structure can be described by a simple formula, which can be input as a line in the program used to analyse the data obtained from an experiment.

3.3 Statisticians at Iowa State University

In parallel with Nelder’s work was the work of Oscar Kempthorne and his colleagues. Kempthorne worked at the Statistics Department at Rothamsted Experimental Station from 1941 to 1946. He spent most of the rest of his career at Iowa State University. While there, he obtained a grant from the Aeronautical Research Laboratory to work with his colleagues on various problems in the design of experiments.

Their technical report [Reference Kempthorne, Zyskind, Addelman, Throckmorton and White26] was completed in November 1961, and consisted of 218 typed pages. It uses the phrases ‘experimental structure’ and ‘response structure’ for what we call ‘block structure’. Sometimes the treatments were also included in this structure. Chapter 3 is based on the PhD theses of Zyskind [Reference Zyskind40] and Throckmorton [Reference Throckmorton36]; part of this was later published as [Reference Zyskind41].

With hindsight, it seems that they were trying to define poset block structures, but they managed to confuse the poset M of coordinates with the lattice of partitions. They denoted the universal partition U by $\mu $ , and the equality partition E by $\varepsilon $ . They used complicated formulae, called symbolic representations, to explain the poset M, but then included $\mu $ and $\varepsilon $ in the corresponding Hasse diagram, which they called the structure diagram. They dealt with all posets of size at most four, and showed $16$ of the $63$ posets of size five.

Figure 5 shows three of their block structures. The first of these is also in Figure 4; the last one cannot be obtained by crossing and nesting, so it needs two formulae.

Figure 5 Some orthogonal block structures in [Reference Kempthorne, Zyskind, Addelman, Throckmorton and White26].

3.4 Unifying the theory

In [Reference Speed, Bailey, Schultz, Praeger and Sullivan34], Speed and Bailey aimed to combine the two approaches by explaining Nelder’s ‘simple orthogonal block structures’ and Throckmorton’s ‘complete balanced block structures’ as ‘association schemes derived from finite distributive lattices of commuting uniform equivalence relations’. They noted that the words ‘permutable’ and ‘permuting’ were sometimes used in place of ‘commuting’. Each partition is defined by a ‘hereditary’ subset of the poset M. This is the dual notion to down-set. A subset H of M is hereditary if, whenever $m\in H$ and $m\sqsubseteq m'$ , then $m'\in H$ . Then $\Omega = \Omega _1 \times \cdots \times \Omega _N$ (where $N=\left |M\right |$ ). Two elements $(\alpha _1, \ldots , \alpha _N)$ and $(\beta _1, \ldots , \beta _N)$ are in the same part of the partition $\Pi _H$ if and only if $\alpha _i=\beta _i$ for all i in H.

To match the partial order on partitions to the partial order $\subseteq $ on subsets of M, they defined $\preccurlyeq $ in the opposite way to what we do here. They proved that every distributive block structure is isomorphic to a poset block structure, but did not use the latter term, even though they showed that the construction depends on a partially ordered set.

They also explained that most of the theory extends to what we now call an orthogonal block structure, where the lattice is modular but not necessarily distributive. Figure 6 shows the corresponding Hasse diagrams in their two examples. In the one on the left, the nontrivial partitions form the rows, columns and the Greek and Latin letter partitions of a pair of mutually orthogonal Latin squares. Note that the underlying set has size $n^2$ with $n \notin \{1,2,6\}$ , since it is well known that there exists a pair of two mutually orthogonal Latin squares of order n if and only if $n\notin \{1, 2, 6\}$ . One way of achieving the one on the right is to use some carefully chosen subgroups of the elementary abelian group of order $16$ .

Figure 6 Hasse diagrams of two nondistributive orthogonal block structures.

In [Reference Bailey and McAvaney2], Bailey restricted attention to distributive block structures, using the term ‘ancestral subset’ in place of ‘hereditary subset’ and drawing the Hasse diagrams in the way consistent with our current use of the refinement partial order $\preccurlyeq $ . This cited [Reference Zyskind40] as well as [Reference Throckmorton36], and commented that Holland [Reference Holland22] ‘defines the automorphism group of a poset block structure to be a generalised wreath product’. The explicit form for such a group was given in [Reference Bailey, Praeger, Rowley and Speed9], following the arguments in [Reference Holland22].

Paper [Reference Bailey, Praeger, Rowley and Speed9] gives a formal definition of poset block structure and an automorphism of such a structure. It shows that, in the finite case, the automorphism group is the generalised wreath product of the relevant symmetric groups. The argument draws on work of Wells [Reference Wells37] for semi-groups. The paper also states that, in the finite case, the generalised wreath product of permutation groups is the same as that constructed by [Reference Holland22, Reference Silcock33].

In [Reference Speed and Bailey35], Speed and Bailey discuss factorial dispersion models, which are statistical models whose underlying structure is a poset block structure. Now hereditary subsets are called filters and the refinement partial order is shown in the same way as we do here.

Papers [Reference Bailey and McAvaney2, Reference Bailey, Praeger, Rowley and Speed9, Reference Speed and Bailey35] have the disadvantage that the partial order on the subsets of M is the wrong way up for inclusion. In the current paper, our use of down-sets rather than hereditary subsets gets round this problem.

In [Reference Houtman and Speed23], Houtman and Speed extend the meaning of ‘orthogonal block structure’ to mean a particular desirable property of covariance matrices. This is even more general than their being based on an association scheme, so we do not use that meaning here.

The survey paper [Reference Bailey4] explains the combinatorial aspects of all these ideas in more detail. It notes that a ‘complete balanced response structure’ is not necessarily a poset block structure, but can always be extended to one by the inclusion of infima.

It also discusses automorphisms. In the present paper, an automorphism of a poset block structure is a permutation of the base-set $\Omega $ which preserves each of the relevant partitions. In [Reference Bailey4, Reference Bailey and Jungnickel8], this is called a ‘strong automorphism’, while a ‘weak automorphism’ preserves the set of these partitions. These are called ‘strict automorphism’ and ‘automorphism’, respectively, in [Reference Cameron11].

If there are nonidentity weak automorphisms, then under suitable conditions we can extend our group by adjoining these. We do not discuss this here, but note that three of the types of primitive group in the celebrated O’Nan–Scott theorem [Reference Scott32] can be realised in this way: affine groups, wreath products with product action, and diagonal groups.

3.5 Statistics and group theory

Why do statisticians care about these groups? First, because of the need to randomise. An experimental design is an allocation of treatments to the elements of the base-set $\Omega $ . To avoid possible bias, this allocation is then randomised by applying a permutation chosen at random from the automorphism group of the block structure. Denote by $Y_\alpha $ the random variable for the response on plot $\alpha $ . The method of randomisation allows us to assume that the covariance of $Y_\alpha $ and $Y_\beta $ is equal to the covariance of $Y_\gamma $ and $Y_\delta $ (but unknown in advance) if and only if $(\gamma ,\delta )$ is in the same orbit of the action of the automorphism group on $\Omega \times \Omega $ as at least one of $(\alpha ,\beta )$ and $(\beta ,\alpha )$ .

For the full generalised wreath product of symmetric groups, these orbits on pairs are precisely the association classes of the association scheme described in [Reference Speed, Bailey, Schultz, Praeger and Sullivan34]. Thus the eigenspaces of the covariance matrix are known in advance of data collection. These eigenspaces are called strata in [Reference Nelder29, Reference Nelder30]. Data can be projected onto each stratum for a straightforward analysis.

Now suppose that each symmetric group $G_i$ in the generalised wreath product is replaced by a subgroup $H_i$ . Lemma 11 in [Reference Bailey, Praeger, Rowley and Speed9] shows that the eigenspaces are known in advance if and only if the permutation character of the generalised wreath product is multiplicity-free (or a slight weakening of this, because the covariance-matrix must be symmetric). In particular, so long as each subgroup $H_i$ is doubly transitive then the strata are the same as they are for the generalised wreath product of symmetric groups.

Paper [Reference Speed, Bailey, Schultz, Praeger and Sullivan34] concludes with acknowledgements to several people, including P. J. Cameron and D. E. Taylor. These two had explained to the authors of [Reference Speed, Bailey, Schultz, Praeger and Sullivan34] the importance of having a permutation character which is multiplicity-free.

4.2 Properties of OB groups

4.2.2 Products

We consider direct and wreath products of transitive groups.

Theorem 4.7. Let G and H be transitive permutation groups. Then $G\wr H$ (in its imprimitive action) has the OB property if and only if G and H do.

Proof. If G and H act on $\Gamma $ and $\Delta $ respectively, then $G\wr H$ acts on $\Gamma \times \Delta $ , and preserves the canonical partition $\Pi _0$ into the sets $\Gamma _\delta =\{(\gamma ,\delta ):\gamma \in \Gamma \}$ for $\delta \in \Delta $ . It was shown in [Reference Anagnostopoulou-Merkouri, Cameron and Suleiman1] that any invariant partition for $G\wr H$ is comparable with $\Pi _0$ ; the partitions below $\Pi _0$ induce a G-invariant partition on each part of $\Pi _0$ , while the partitions above $\Pi _0$ induce an H-invariant partition on the set of parts.

Suppose that G and H have the OB property, and let $\Sigma _1$ and $\Sigma _2$ be $G\wr H$ -invariant partitions. If one is below $\Pi _0$ and the other above, then they are comparable, and so they commute. If both are below, then they commute since G has the OB property; and if both are above, then they commute since H has the OB property. So the OBS is obtained by nesting the OBS for G in that for H.

Conversely, suppose that $G\wr H$ has the OB property. Then the partitions below $\Pi _0$ commute, so G has the OB property; and the partitions above $\Pi _0$ commute, so H has the OB property.

Corollary 4.8. Let G and H be permutation groups. If $G\times H$ has the OB property in its product action then G and H both have the OB property.

Proof. As in [Reference Anagnostopoulou-Merkouri, Cameron and Suleiman1], $G\times H$ is a subgroup of $G\wr H$ . So, if $G\times H$ has the OB property, then $G\wr H$ has the OB property by Proposition 4.6, and the result holds by Theorem 4.7.

We will see later (after Theorem 4.14) that the converse is false. However, we have some positive results.

First we prove some general facts about invariant partitions of direct products of an arbitrary number of groups in their product action, and slightly extend a result in [Reference Anagnostopoulou-Merkouri, Cameron and Suleiman1], proving that the direct product of an arbitrary number of primitive groups in its product action is pre-primitive. This result is interesting in its own right, but it will also be used to show that a generalised wreath product of primitive groups is pre-primitive, which constitutes a part of Theorem 4.19. First we give some language to describe partitions of products.

Let G and H act transitively on $\Gamma $ and $\Delta $ respectively, and let $\Pi $ be a $(G\times H)$ -invariant partition of $\Gamma \times \Delta $ . We define two partitions of $\Gamma $ in the following way:

• ○ Let P be a part of $\Pi $ . Let $P_0$ be the subset of $\Gamma $ defined by

  $$\begin{align*}P_0 = \{\gamma \in \Gamma \, : \, (\exists \delta\in \Delta)((\gamma, \delta)\in P)\}. \end{align*}$$

  We claim that the sets $P_0$ arising in this way are pairwise disjoint. For suppose that $\gamma \in P_0\cap Q_0$ , where $Q_0$ is defined similarly for another part Q of $\Pi $ ; suppose that $(\gamma , \delta _1)\in P$ and $(\gamma , \delta _2)\in Q$ . There is an element $h\in H$ mapping $\delta _1$ to $\delta _2$ . Then $(1, h)$ maps $(\gamma , \delta _1)$ to $(\gamma , \delta _2)$ , and hence maps P to Q, and $P_0$ to $Q_0$ ; but this element acts trivially on $\Gamma $ , so $P_0 = Q_0$ . It follows that the sets $P_0$ arising in this way form a partition of $\Gamma $ , which we call the G-projection partition of $\Gamma $ induced by $\Pi $ .

• ○ Choose a fixed $\delta \in \Delta $ , and consider the intersections of the parts of $\Pi $ with $\Gamma \times \{\delta \}$ . These form a partition of $\Gamma \times \{\delta \}$ and so, by ignoring the second factor, we obtain a partition of $\Gamma $ called the G-fibre partition of $\Gamma $ induced by $\Pi $ . Now the action of the group $\{1\}\times H$ shows that it is independent of the element $\delta \in \Delta $ chosen.

We note that the G-projection partition and the G-fibre partition are both G-invariant, and the second is a refinement of the first. In a similar way we get H-fibre and H-projection partitions of $\Delta $ , both H-invariant.

Proposition 4.9. Let $\Pi $ be a $G\times H$ -invariant partition of $\Gamma \times \Delta $ , where G and H act transitively on $\Gamma $ and $\Delta $ respectively. Then the projection and fibre partitions of $\Pi $ on $\Gamma $ are equal if and only if $\Pi $ is obtained by crossing a G-invariant partition of $\Gamma $ with an H-invariant partition of $\Delta $ .

Proof. First we observe that the projection and fibre partitions on $\Gamma $ agree if and only if those on $\Delta $ agree. For the pairs in a part P of $\Pi $ are the edges of a bipartite graph on $A\cup B$ , where A and B are parts of the projection partitions on $\Gamma $ and $\Delta $ respectively; the valency of a point in A is equal to the number of points of B in a part of the fibre partition on $\Delta $ , which we will denote by a; and similarly the valencies b of the points in B. Then counting edges of the graph (that is, pairs in part P of $\Pi $ ), we see that $|A|a=|B|b$ . Now the fibre and projection partitions on $\Gamma $ agree if and only if $|A|=b$ , which is equivalent to $|B|=a$ .

Moreover, if this equality holds, then every pair in $A\times B$ lies in the same part of $\Pi $ , so A and B are parts of both the projection and fibre partitions on the relevant sets. In this case, $\Pi $ is obtained by crossing these partitions.

Conversely, it is easy to see that if $\Pi $ is obtained by crossing, then the fibre and projection partitions coincide.

Next we introduce the notion of partition orthogonality.

Definition

Let $G, H$ be transitive permutation groups, on $\Gamma $ , $\Delta $ respectively, as above. We say that G and H are partition-orthogonal if the only $G\times H$ -invariant partitions of $\Gamma \times \Delta $ are of the form $\{\Gamma _i\times \Delta _j \mid i\in \{1, \ldots , m\}, j\in \{1, \ldots , n\}\}$ where $\{\Gamma _1, \ldots , \Gamma _m\}$ is a G-invariant partition of $\Gamma $ and $\{\Delta _1, \ldots , \Delta _n\}$ is an H-invariant partition of $\Delta $ .

Lemma 4.10. Let $G_i \leq \mathop {\mathrm {Sym}}(\Omega _i)$ for $i\in \{1, \ldots , m\}$ be transitive, and let $G = G_1\times \cdots \times G_m$ act on $\Omega = \Omega _1 \times \cdots \times \Omega _m$ component-wise. If $G_i$ and $G_j$ are partition-orthogonal for all $i, j\in \{1, \ldots , m\}$ with $i\ne j$ , then the G-invariant partitions are precisely the products of $G_i$ -invariant partitions for $i\in \{1, \ldots , m\}$ .

Proof. We prove the claim by induction. If $m = 2$ , then the claim follows by the definition. Suppose that the claim holds for $m - 1$ factors. Let $H = G_1\times \cdots \times G_{m - 1}$ and suppose for a contradiction that there is some G-invariant partition $\Pi $ which is not a direct product of partitions of the sets $\Omega _i$ . Then the H-fibre and H-projection partitions induced on $\Omega _1\times \cdots \times \Omega _{m - 1}$ by $\Pi $ must differ.

By the induction hypothesis, all the H-invariant partitions are direct products of partitions, and therefore there must exist some $i\in \{1, \ldots , m - 1\}$ such that the $G_i$ -fibre and the $G_i$ -projection partition induced on $\Omega _i$ by $\Pi $ differ. However, this means that the partition induced on $G_i\times G_m$ is not a direct product of partitions of $\Omega _i\times \Omega _m$ , which is a contradiction since we have assumed that $G_i$ and $G_m$ are partition-orthogonal.

Therefore, every G-invariant partition of $\Omega $ must be a direct product of partitions of the sets $\Omega _i$ .

Lemma 4.11. Let $G_1 \leq \mathop {\mathrm {Sym}}(\Omega _1), \ldots , G_m\leq \mathop {\mathrm {Sym}}(\Omega _m), H\leq \mathop {\mathrm {Sym}}(\Delta )$ be transitive groups. If H is partition-orthogonal to $G_i$ for all $i\in \{1, \ldots , m\}$ , then H is partition-orthogonal to $G_1\times \cdots \times G_m$ .

Proof. Let $G = G_1\times \cdots \times G_m \times H$ and $\Omega = \Omega _1\times \cdots \times \Omega _m \times \Delta $ . We prove the claim by induction on m.

We first prove the claim for $m = 2$ . Let $(\alpha , \beta , \delta ), (\alpha ', \beta ', \delta ')\in \Omega _1\times \Omega _2\times \Delta $ . First consider the H-fibre partition $\Pi _{\Delta }$ of $\Delta $ induced by $\Pi $ . Note that by definition $(\alpha , \beta , \delta )$ and $(\alpha , \beta , \delta ')$ are in the same part of $\Pi $ if and only if $\delta $ and $\delta '$ are in the same part of $\Pi _{\Delta }$ .

Now consider the $(G_1\times H)$ -fibre partition $\Pi _{\Omega _1\times \Delta }$ of $\Omega _1\times \Delta $ induced by $\Pi $ . Since $G_1$ and H are partition-orthogonal by assumption, we must have $\Pi _{\Omega _1\times \Delta } = \Pi _{\Omega _1}\times \Pi _{\Delta }$ , where $\Pi _{\Omega _1}$ denotes the $G_1$ -fibre partition of $\Omega _1$ induced by $\Pi $ . This means that $(\alpha , \beta , \delta )$ and $(\alpha ', \beta , \delta ')$ are in the same part of $\Pi $ if and only if $\alpha $ and $\alpha '$ are in the same part of $\Pi _{\Omega _1}$ and $\delta $ and $\delta '$ are in the same part of $\Pi _{\Delta }$ .

An entirely similar argument shows that $(\alpha , \beta , \delta )$ and $(\alpha , \beta ', \delta ')$ are in the same part of $\Pi $ if and only if $\beta $ and $\beta '$ are in the same part of the $G_2$ -fibre partition $\Pi _{\Omega _2}$ of $\Omega _2$ induced by $\Pi $ and $\delta $ and $\delta '$ are in the same part of $\Pi _{\Delta }$ .

It therefore follows that $(\alpha , \beta ,\delta )$ and $(\alpha ', \beta ', \delta ')$ are in the same part of $\Pi $ if and only if $\alpha $ and $\alpha '$ are in the same part of $\Pi _{\Omega _1}$ , $\beta $ and $\beta '$ are in the same part of $\Pi _{\Omega _2}$ and $\delta $ and $\delta '$ are in the same part of $\Pi _{\Delta }$ , and so

$$\begin{align*}\Pi = \Pi_{\Omega_1}\times \Pi_{\Omega_2}\times \Pi_{\Delta} \end{align*}$$

which proves the claim.

Now suppose that the claim holds for all integers less than m. Then, it follows that H is partition-orthogonal to $G_1\times \cdots \times G_{m - 1}$ . Now, since H is partition-orthogonal to both $G_1\times \cdots \times G_{m -1}$ and $G_m$ , using the inductive hypothesis once more gives us that H is indeed partition-orthogonal to $G_1\times \cdots \times G_m$ .

Lemma 4.12. Let $G\leq \mathop {\mathrm {Sym}}(\Gamma )$ and $H\leq \mathop {\mathrm {Sym}}(\Delta )$ be partition-orthogonal pre-primitive groups. Then $G\times H$ in its product action is pre-primitive.

Proof. Let $\Pi $ be a $G\times H$ -invariant partition of $\Gamma \times \Delta $ . Since G and H are partition-orthogonal, $\Pi $ is the direct product of a G-invariant partition $\Pi _G$ and an H-invariant partition $\Pi _H$ . Since both G and H are pre-primitive, it follows that $\Pi _G$ and $\Pi _H$ are orbit partitions of some subgroups $G^*$ and $H^*$ of G and H respectively. It is then easy to check that $\Pi $ is the orbit partition of $G^*\times H^*$ , which proves the claim.

Theorem 4.13. Let $G_i \leq \mathop {\mathrm {Sym}}(\Omega _i)$ for $i\in \{1, \ldots , m\}$ , and let $G_i$ act primitively on $\Omega _i$ for all $i\in \{1, \ldots , m\}$ . Then $G = G_1\times \cdots \times G_m$ in its product action is pre-primitive.

Proof. Abelian primitive groups are cyclic of prime order. So, by rearranging the components if necessary, we can write G as a direct product of elementary abelian groups of different prime power order and nonabelian primitive groups.

It has been shown in [Reference Anagnostopoulou-Merkouri, Cameron and Suleiman1] that two primitive groups are partition-orthogonal if and only if they are not cyclic of the same prime order. Therefore, if P and Q are two elementary abelian groups of orders $p^a$ and $q^b$ respectively, with $p\ne q$ , then it follows by Lemma 4.11 that every component of Q is partition-orthogonal to P, and then applying Lemma 4.11 again, we get that P must be partition-orthogonal to Q. Similarly, we get that any elementary abelian group and any nonabelian primitive group are partition-orthogonal. Then Lemma 4.10 gives us that G can be written as a direct product of mutually partition-orthogonal factors, and it is hence pre-primitive by Lemma 4.12.

4.2.3 Regular groups

It follows from Corollary 4.2 that, if G is a regular permutation group, then G has the OB property if and only if any two subgroups of G commute. These groups were determined by Iwasawa [Reference Iwasawa25]; we refer to Schmidt [Reference Schmidt31, Chapter 2] for all the material we require. In this section we use the term quasi-hamiltonian, taken from [Reference Chataut, Dixon, Baginski, Fine and Gaglioni15], for a group in which any two subgroups commute. (The term will not be used outside this section.)

We warn the reader that both Iwasawa and Schmidt consider hypotheses which are more general in two ways:

• ○ they consider groups whose subgroup lattices are modular, which is weaker than requiring all subgroups to commute;

• ○ they consider infinite as well as finite groups.

We have not found a reference for precisely what we want, so we give a direct proof of the first part; the second is [Reference Schmidt31, Theorem 2.3.1].

Theorem 4.14. The following are true:

1. (a) A finite group G is quasi-hamiltonian if and only if it is the direct product of quasi-hamiltonian subgroups of prime power order.

2. (b) Suppose that p is prime, and G is a nonabelian quasi-hamiltonian p-group. Then either

   • ○ $G=Q_8\times V$ , where $Q_8$ is the quaternion group of order $8$ and V an elementary abelian $2$ -group; or

   • ○ G has an abelian normal subgroup A with cyclic factor group and there is $b\in G$ with $G=A\langle b\rangle $ and s such that $b^{-1}ab=a^{1+p^s}$ for all $a\in A$ , with $s\ge 2$ if $p=2$ .

Here is the proof of part (a). Suppose that $P_1$ and $P_2$ are Sylow p-subgroups of the quasi-hamiltonian group G. Then $P_1P_2$ is a subgroup, and $|P_1P_2|=|P_1|\cdot |P_2|/|P_1\cap P_2|$ . Since $P_1$ and $P_2$ are Sylow subgroups, this implies that $P_1=P_2$ . So all Sylow subgroups of G are normal, and G is nilpotent. Thus it is the direct product of its Sylow subgroups. Since quasi-hamiltonicity is clearly inherited by subgroups, the result follows

Conversely, if G is nilpotent with quasi-hamiltonian Sylow subgroups, then any subgroup is nilpotent and hence a direct product of its Sylow subgroups. Factors whose orders are powers of different primes commute; factors whose orders are powers of the same prime commute by hypothesis. So any two subgroups commute.

Note that not every quasi-hamiltonian group is a Dedekind group, namely a group all of whose subgroups are normal; so the OB property lies strictly between transitivity and pre-primitivity. Note also that $Q_8$ (acting regularly) is quasi-hamiltonian but $Q_8\times Q_8$ in the product action is not; so the OB property is not closed under direct product.

For groups with a regular normal subgroup, we have the following result.

Theorem 4.15. If $G\leq \mathop {\mathrm {Sym}}(\Omega )$ is a transitive group containing a regular normal subgroup N, then G is OB if and only if the subgroups of N normalised by $G_{\alpha }$ commute.

Proof. Suppose that G is OB. Since N is a regular normal subgroup of G we can write $G = NG_{\alpha }$ for some $\alpha \in \Omega $ , where $N\cap G_{\alpha } = 1$ and we can identify $\Omega $ with N in such a way that $G_{\alpha }$ acts by conjugation and N acts by right multiplication.

We first show that the subgroups containing $G_{\alpha }$ are of the form $HG_{\alpha }$ for some $H\leq N$ invariant under the action of $G_{\alpha }$ . Let F be a subgroup containing $G_{\alpha }$ . Since $F\leq G = NG_{\alpha }$ all the elements of F are of the form $ng$ where $n\in N$ and $g\in G_{\alpha }$ . Then since $G_{\alpha }\leq F$ it follows that $n = ngg^{-1}\in F$ . Hence, $F = HG_{\alpha }$ , where $H = N\cap F \leq N$ . Since both F and N are invariant under the action of $G_{\alpha }$ , so is H.

Let $KG_{\alpha }$ , $LG_{\alpha }$ be two such subgroups. Since G is OB, they commute (Corollary 4.2), and we have

(4.1) $$ \begin{align} KG_{\alpha}LG_{\alpha} = LG_{\alpha}KG_{\alpha}. \end{align} $$

But $G_{\alpha }L = LG_{\alpha }$ and $G_{\alpha }K = KG_{\alpha }$ since $KG_{\alpha }, LG_{\alpha } \leq G$ . Therefore, by Equation (4.1) we get

$$\begin{align*}KLG_{\alpha} = LKG_{\alpha} \end{align*}$$

and intersecting both sides with N gives us $KL = LK$ . Since $K, L$ were arbitrary $G_{\alpha }$ -invariant subgroups of N the claim holds.

Conversely, suppose that all the $G_{\alpha }$ -invariant subgroups of N commute and consider subgroups $KG_{\alpha }, LG_{\alpha }\leq G$ , where K and L are subgroups of N normalised by $G_\alpha $ . Then

$$\begin{align*}KG_{\alpha}LG_{\alpha} = KLG_{\alpha} = LKG_{\alpha} = LG_{\alpha}KG_{\alpha}, \end{align*}$$

and so G is OB (again by Corollary 4.2).

4.2.4 Modularity and distributivity

We have seen at the start of Section 4.2.3 that the subgroup lattice of a group, which clearly determines modularity, does not determine whether the subgroups commute. So we cannot expect a characterisation of the OB property in terms of the lattice of subgroups containing a given point stabiliser. But is there anything to say here?

An example of a transitive group in which the lattice of invariant equivalence relations is the pentagon ( $P_5$ in Figure 2) is the following. Let G be the 2-dimensional affine group over a finite field F of order q, and let G act on the set of flags (incident point-line pairs) in the affine plane. The three nontrivial G-invariant relations are ‘same line’, ‘parallel lines’, and ‘same point’. Clearly the equivalence relations ‘same point’ and ‘same line’ do not commute.

Since modularity does not imply the OB property, we could ask whether a stronger property does. We saw in Corollary 4.4 that the property of being a chain does suffice. Is there a weaker property?

Proposition 4.16. Let G be a finite regular permutation group. Then the lattice of G-invariant partitions is distributive if and only if G is cyclic.

This is true because a group with distributive subgroup lattice is locally cyclic, by Ore’s theorem [Reference Schmidt31, Section 1.2], and a finite locally cyclic group is cyclic. Since a cyclic group is Dedekind, it is pre-primitive and so has the OB property.

However, there is no general result along these lines. Even if we assume that the lattice of G-invariant partitions is a Boolean lattice (isomorphic to the lattice of subsets of a finite set), the group may fail to have the OB property, as the next example shows.

Example

Let $G=\mathrm {GL}(n,q)$ acting on the set of maximal chains of nontrivial proper subspaces

$$\begin{align*}V_1<V_2<\cdots<V_{n-1}\end{align*}$$

in the vector space $V=\mathrm {GF}(q)^n$ , where $\dim (V_k)=k$ for $0<k<n$ . The stabiliser B of such a chain is a Borel subgroup of G; if we take $V_k$ to be spanned by the first k basis vectors, then B is the group of upper triangular matrices with nonzero entries on the diagonal. From the theory of algebraic groups, it is known that the only subgroups of G containing B are the parabolic subgroups, the stabilisers of subsets of $\{V_1,\ldots ,V_{n-1}\}$ (see, e.g., [Reference Humphreys24] for the theory). Hence the lattice of G-invariant partitions is isomorphic to the Boolean lattice $B_{n-1}$ of subsets of $\{1,\ldots ,n-1\}$ (the isomorphism reverses the order since the stabiliser of a smaller set of subspaces is larger).

However, the equivalence relations do not all commute. Consider the relations $\Pi _1$ and $\Pi _2$ corresponding to the subgroups fixing $V_1$ and $V_2$ . Thus, two chains are in the relation $\Pi _1$ if they contain the same 1-dimensional subspace, and similarly for $\Pi _2$ . Now starting from the chain $(V_1,V_2,\ldots ,V_n)$ , a move in a part of $\Pi _2$ fixes $V_2$ and moves $V_1$ to a subspace $V_1'$ of $V_2$ ; then a move in $\Pi _1$ fixes $V_1'$ , so the resulting chain begins with a subspace of $V_2$ . But if we move in a part of $\Pi _1$ , we can shift $V_2$ to a different 2-dimensional subspace, and then a move in a part of $\Pi _2$ can take $V_1$ to a subspace not contained in $V_2$ . So $\Pi _1\circ \Pi _2\ne \Pi _2\circ \Pi _1$ , and the lattice is not an OBS.

So G does not have the OB property, even though the lattice of G-invariant partitions is a Boolean lattice (and hence distributive).

4.3 Generalised wreath products

In this section, we prove two main results. The first describes the group-theoretic structure of a generalised wreath product, and will be needed later. The second investigates properties of the generalised wreath product of primitive groups; in particular, they are pre-primitive and hence have the OB property, and we give necessary and sufficient conditions for them to have the PB property.

4.3.1 A group-theoretic result

First we prove a result about generalised wreath products which will be needed later.

We note that, if p is a minimal element of a poset M, then $\{p\}$ is a down-set, and so corresponds to a partition $\Pi $ of the domain $\Omega $ of the generalised wreath product of a family of groups over M.

Theorem 4.17. Let G be the generalised wreath product of the groups $G(m)$ over a poset M, acting on a set $\Omega $ . Let p be a minimal element of M. Let $\Pi $ be the corresponding partition of $\Omega $ , H the group induced on the set of parts by G, N the stabiliser of all parts of $\Pi $ . Then

1. (a) H is isomorphic to the generalised wreath product of the groups $G(q)$ for $q\in M\setminus \{p\}$ ;

2. (b) N is a direct product of copies of $G(p)$ , where there is an equivalence relation $\sim $ on the set of parts of $\Pi $ (determined by the poset M) such that each direct factor acts in the same way on the parts in one equivalence class and fixes every point in the other parts;

3. (c) G is a semidirect product $N\rtimes H$ .

Proof. For (a), we note that, since p is minimal, suppressing the pth coordinate of every tuple in $\Omega $ gives the generalised wreath product of the remaining groups indexed by the elements different from p.

Part (b) is proved using the definition of a generalised wreath product. The equivalence relation is defined as follows: for parts P and Q of $\Pi $ , $P\sim Q$ if and only if P and Q lie in the same part of $\Pi \vee \Phi $ for all partitions $\Phi $ of the poset block structure defined by M which are incomparable to $\Pi $ .

First note that since N fixes the parts of $\Pi $ , it must also fix the parts of every partition lying above $\Pi $ . Therefore, only parts of partitions incomparable to $\Pi $ can be moved by N. Now let $\Phi $ denote a partition incomparable to $\Pi $ . Note that since $\Pi \preccurlyeq \Pi \vee \Phi $ , the parts of $\Phi $ contained in the same part of $\Pi \vee \Phi $ can only be permuted amongst themselves by N. Hence, if the actions of $h\in N$ on two parts of $\Pi $ are equivalent, then those two parts must be contained in the same part of $\Pi \vee \Phi $ for every partition $\Phi $ of $\Omega $ incomparable to $\Pi $ .

It now remains to show that if $P,Q\in \Pi $ are such that $P\sim Q$ , then N acts in the same way on P and Q. Let $\gamma , \delta $ lie in P and Q, and moreover suppose that they are contained in the same part of $\Phi $ for every partition $\Phi $ of $\Omega $ incomparable to $\Pi $ . It suffices to show that every $h\in N$ maps $\gamma $ and $\delta $ to the same part of $\Phi $ for every $\Phi $ incomparable to $\Pi $ .

Now h can be written as a product $\prod _{\Phi } h_{\Phi }$ , where each factor $h_{\Phi }$ encodes the permutation induced by h of the parts of the corresponding partition $\Phi $ of $\Omega $ induced by h. Hence, it suffices to show that $h_{\Phi }$ maps $\gamma $ and $\delta $ to the same part for an arbitrary partition $\Phi $ of $\Omega $ incomparable to $\Pi $ . We may assume without loss of generality that $\Phi $ is join-indecomposable, since every element is a join of JI elements, and the distributive law implies that if a collection of JI elements are incomparable with $\Pi $ then so is their join.

Let m be the element corresponding to $\Phi $ in the poset M. Using the notation established in [Reference Bailey, Praeger, Rowley and Speed9], we note that $\gamma $ and $\delta $ must be such that $\gamma _i = \delta _i$ for all $i\sqsupseteq m$ in M. Therefore,

$$\begin{align*}(\gamma h_{\Phi})_i = \gamma_i(\gamma \pi^i (h_{\Phi})_i) = \delta_i(\delta \pi^i (h_{\Phi})_i) = (\delta h_{\Phi})_i \end{align*}$$

for all $i\sqsupseteq m$ , which proves the claim.

We finally note that $\sim $ is dependent only on the poset M and not the group G.

For (c), we have to show that H normalises N and that the action of H extends to $\Omega $ . The first statement is clear since N is the subgroup fixing all parts of $\Pi $ . For the second, note that H acts on the set of $(|M|-1)$ -tuples; extend each element to act on $|M|$ -tuples by acting as the identity on the p-th coordinate.

4.3.2 Generalised wreath products of primitive groups

In this section, we will use the notation for poset block structures and generalised wreath products defined in Section 2.5. Moreover, let $[N]$ denote the set $\{1, \ldots , N\}$ , and for every subset J of M, let $X_J$ be the index set of J, namely $\{i\in [N] \, : \, m_i\in J\}$ . We then define $P_J$ to be the partition of $\Omega $ whose set of parts is

$$\begin{align*}\left\{\prod_{j \in X_J} \Omega_j \times \prod_{k\in [N]\setminus X_J} \{\alpha_k\} : \alpha_k \in \Omega_k \text{ for all } k\in [N]\setminus X_J\right\}. \end{align*}$$

We now prove a small lemma that will be used in the proof of Theorem 4.19.

Lemma 4.18. Let G be the generalised wreath product of the groups $G(m_i)$ over the poset M. Let $J, K$ be down-sets of M such that $P_K \preccurlyeq P_J$ , let $\Gamma $ be a part of $P_{J}$ , and let $\Delta $ be the set of parts of $P_K$ contained in $\Gamma $ . Then the permutation group $G(\Delta ,\Gamma )$ induced by the setwise stabiliser of $\Gamma $ on $\Delta $ is isomorphic to the generalised wreath product of the groups $G(m_i)$ for $i\in X_J\setminus X_K$ .

Proof. Let $\Gamma = \prod _{j \in X_J} \Omega _j \times \prod _{i\in [N]\setminus X_J} \{\alpha _i\}$ , where $\alpha _i$ is fixed for $i\in [N]\setminus X_J$ . Note that the setwise stabiliser $G_\Gamma $ in G must be equal to the generalised wreath product of the groups $H(m_i)$ , where $H(m_i) = G(m_i)$ for all $i\in X_J$ and $H(m_i) = (G(m_i))_{\alpha _i}$ for $i\not \in X_J$ . Now since the elements of $\Delta $ are blocks of imprimitivity of G, they are also blocks of $G_\Gamma $ , and moreover, since $\Gamma $ is a block of $P_J$ , it follows that $G_\Gamma $ induces a permutation group on $\Delta $ . Let $\rho $ denote the associated permutation representation.

Note that every element of $\Delta $ is of the form $\prod _{i\in X_K}\Omega _i \times \prod _{i\in [N]\setminus X_K}\{\alpha _i\}$ , where $\alpha _i$ is fixed for $i\in X_J\setminus X_K$ . Therefore, $\ker \rho $ must fix all elements of $\Omega _i$ for $i\in X_J\setminus X_K$ , must fix $\alpha _i$ for $i\in [N]\setminus X_J$ , and can permute the elements of $\Omega _i$ for $i\in X_K$ in any way $G_\Gamma $ allows. Hence, $\ker \rho $ is equal to the generalised wreath product of $L(m_i)$ , where $L(m_i) = G(m_i)$ for $i\in X_K$ , $L(m_i) = 1$ for $i\in X_J\setminus X_K$ , and $L(m_i) = G(m_i)_{\alpha _i}$ for $i\in [N]\setminus X_J$ . We then deduce that

$$\begin{align*}G(\Delta,\Gamma) \cong G_\Gamma/ \ker \rho, \end{align*}$$

the generalised wreath product of $G(m_i)$ for $i\in X_J\setminus X_K$ , as claimed.

We are now in a position to state and prove the main theorem of this section.

Theorem 4.19. If $G(m_i)$ is primitive for every $i\in [N]$ , then the following hold for their generalised wreath product G:

1. (a) G is pre-primitive, and hence has the OB property;

2. (b) the following are equivalent:

   1. (i) G has the PB property;

   2. (ii) the only G-invariant partitions are the ones corresponding to down-sets in M;

   3. (iii) there do not exist incomparable elements $m_i,m_j\in M$ such that $G(m_i)$ and $G(m_j)$ are cyclic groups of the same prime order.

Proof. Let K denote the direct product of the groups $G(m_i)$ (for $i\in [N]$ ) in its product action and P denote the lattice of partitions corresponding to down-sets in M.

(a) Since pre-primitivity is upward-closed, it suffices to show that K can be embedded in G. Then the claim will follow by Theorem 4.13. Let H be the set of all functions $f = \prod _{i\in [N]} f_i \in G$ such that $f_i$ sends all elements of $\Omega ^i$ to the same element of $G(m_i)$ for all $i\in [N]$ . We will show that H is permutation isomorphic to K.

We first start by showing that $H\leq G$ . To prove closure, it suffices to show that for $f, h\in H$ , then $(fh)_i$ sends all elements of $\Omega ^i$ to the same element of $G(m_i)$ for every $i\in [N]$ . We can do this by showing that if $\gamma , \delta \in \Omega $ , then $fh$ acts on both $\gamma $ and $\delta $ with the same group element on each coordinate. We will slightly abuse notation and for $f = \prod _{i\in [N]} f_i\in H$ , we will write $\mathop {\mathrm {im}}(f_i)$ for the element that $f_i$ maps all the elements of $\Omega ^i$ to, instead of the set containing just this element. Now let $g = \mathop {\mathrm {im}}(f_i)$ and $g' = \mathop {\mathrm {im}}(h_i)$ , then

$$\begin{align*}(\gamma fh)_i = (\gamma f)_i(\gamma f\pi^ih_i) = \gamma_i(\gamma\pi^if_i)(\gamma f\pi^ih_i) = \gamma_igg' \end{align*}$$

for all $i\in [N]$ . Similarly,

$$\begin{align*}(\delta fh)_i = (\delta f)_i(\delta f\pi^ih_i) = \delta_i(\delta\pi^if_i)(\delta f\pi^ih_i) = \delta_igg' \end{align*}$$

for all $i\in [N]$ , and therefore, $fh\in H$ . For i in $[N]$ , let $z_i$ be the function which maps all the elements of $\Omega ^i$ to the inverse of $\mathrm {im}(f_i)$ . Put $z = \prod _{i\in [N]}z_i$ . Then

$$\begin{align*}(\gamma fz)_i = (\gamma f)_i(\gamma f\pi^iz_i) = \gamma_i(\gamma\pi^if_i)(\gamma f\pi^iz_i) = \gamma_igg^{-1} = \gamma_i \end{align*}$$

for all $i\in [N]$ , and thus $z = f^{-1}\in H$ .

We now need to show that H is permutation isomorphic to K in its product action on $\Omega $ . Let $\phi : G \to K$ be the function defined by the formula $(\prod _{i\in [N]}f_i) \phi = \prod _{i\in [N]} \mathop {\mathrm {im}}(f_i)$ , and let $\mathop {\mathrm {id}}$ denote the identity function. Note that $\phi $ is clearly a bijection by construction, and also

$$\begin{align*}(\delta f)_i = \delta_ig_i \end{align*}$$

for all $i\in [N]$ , where $g_i=\mathop {\mathrm {im}}(f_i)$ , and hence

$$\begin{align*}(\delta f) = \delta \left(\prod_{i\in[N]} \mathop{\mathrm{im}}(f_i)\right) = \delta (f \phi) = (\delta \mathop{\mathrm{id}})(f\phi) ,\end{align*}$$

which completes the proof of (a).

We now prove (b).

(ii) $\Rightarrow $ (i). Note that if the only partitions preserved by G are the ones in P, then clearly G is PB.

(iii) $\Rightarrow $ (ii). We prove the contrapositive. Suppose that there are further partitions fixed by G other than the ones corresponding to down-sets in M, and let $\Pi $ be such a partition. Since $K\leq G$ , it follows that $\Pi $ is also preserved by K. Therefore, by the arguments in the proof of Theorem 4.13 we deduce that since all of the $G(m_i)$ s are primitive, one of the following must hold:

• ○ All the $G(m_i)$ s are mutually partition-orthogonal, and so $\Pi $ is of the form

  $$\begin{align*}(\alpha_1, \alpha_2, \ldots, \alpha_n) \sim_J (\beta_1, \beta_2, \ldots, \beta_n) \iff (\forall i\not\in X_J)(\alpha_i = \beta_i), \end{align*}$$

  where J is not a down-set of M;

• ○ at least two of the $G(m_i)$ s, say $G(m_i)$ and $G(m_j)$ , are cyclic of the same prime order, and $\Pi $ is a partition whose corresponding $G(m_i)$ and $G(m_j)$ -fibre partitions are different from the $G(m_i)$ and $G(m_j)$ -projection partitions respectively. Since the degree of $G(m_i)$ and $G(m_j)$ is prime, this can only happen if the fibre partitions are the partitions into singletons and the projection partitions are the partitions into a single part.

If $\Pi $ is of the first type, then there exist some $i, j\in [N]$ such that $m_i \sqsubset m_j$ and $m_j\in J$ , but $m_i\not \in J$ . Since $m_i$ and $m_j$ are comparable, there exists a chain $(m_i = a_0, a_1, \ldots , a_k = m_j)$ in M. Thus, $\Pi $ must be preserved by the wreath product $G(a_0)\wr G(a_1)\wr \ldots \wr G(a_k)$ . However, we know that an imprimitive iterated wreath product cannot preserve partitions of equivalence relations where

$$\begin{align*}(\alpha_{a_0}, \ldots, \alpha_{a_k}) \sim (\beta_{a_0}, \ldots, \beta_{a_k}), \end{align*}$$

with $\alpha _{a_s} = \beta _{a_s}$ but $\alpha _{a_t} \neq \beta _{a_t}$ for some $s, t$ such that $s < t$ , because for every $l\in \{1, \ldots , k\}$ , the group $G(a_l)$ permutes whole copies of $\Omega _r$ for each $r \in \{0, \ldots , l-1\}$ .

Hence, $\Pi $ must be of the second type and thus there exist $G(m_i)$ and $G(m_j)$ cyclic of the same prime order and $\Pi $ is a partition whose corresponding $G(m_i)$ and $G(m_j)$ -fibre partitions are the partitions into singletons and the $G(m_i)$ and $G(m_j)$ -projection partitions are the partitions into a single part. If $m_i$ and $m_j$ are related, say $m_i \sqsubset m_j$ then, as above, $\Pi $ must be preserved by the iterated wreath product $G(a_0)\wr G(a_1)\wr \ldots \wr G(a_k)$ . However, knowing what partitions imprimitive iterated wreath products preserve, we deduce that $G(a_0)\wr G(a_1)\wr \ldots \wr G(a_k)$ cannot preserve $\Pi $ and therefore $m_i$ and $m_j$ must be incomparable.

(i) $\Rightarrow $ (iii). We again prove the contrapositive. So suppose that there are two incomparable elements in M, say $m_1$ and $m_2$ , such that the corresponding groups are cyclic of the same prime order p. As defined in Section 2.5,

$$\begin{align*}D(i)=\{m\in M:m\sqsubset m_i\}\end{align*}$$

for $i=1,2$ . Set

$$\begin{align*}S=D(1)\cup D(2),\quad Q=S\setminus\{m_1\}, \quad R=S\setminus\{m_2\},\text{ and } T=Q\cap R. \end{align*}$$

These four sets are all down-sets, and the interval between T and S has the group $G(m_1)\times G(m_2)$ acting, and so we can find partitions fixed by the group, other than the ones corresponding to down-sets in M. More precisely, there are $p + 1$ partitions corresponding to orbit partitions of the diagonal subgroups of $G(m_1)\times G(m_2)$ , and thus preserved by $G(m_1) \times G(m_2)$ . If Y is one of those, then $S, T, Q, R, Y$ form a $N_3$ sublattice (Figure 2) of the invariant partition lattice of G, and hence G fails the PB property. This proves the claim.

4.4 The embedding theorem

The Krasner–Kaloujnine theorem [Reference Krasner and Kaloujnine27] says that, if G is a transitive but imprimitive permutation group on $\Omega $ , then G is embeddable in the wreath product of two groups which can be extracted from G (the stabiliser of a block acting on the block, and G acting on the set of blocks).

In this section, we extend this result to transitive groups which preserve a poset block structure (a distributive lattice of commuting equivalence relations). In particular, our result holds for groups with the PB property. As explained in Subsection 2.5, such a lattice $\Lambda $ is associated with a poset M (so that M consists of the non-E join-indecomposable elements of $\Lambda $ , and $\Lambda $ consists of the down-sets in M). We want to associate a group with each element $m\in M$ such that G is embedded in the generalised wreath product of these groups over the poset M.

Our first attempt was as follows. Take $m\in M$ ; it corresponds to a join-indecomposable partition $\Pi \in \Lambda $ . The join-indecomposability of $\Pi $ implies that there is a unique partition $\Pi ^-$ in $\Lambda $ which is maximal with respect to being below $\Pi $ . Then let $G(m)$ be the permutation group induced by the stabiliser of a part of $\Pi $ acting on the set of parts of $\Pi ^-$ it contains.

However, this does not work. Take G to be the symmetric group $S_6$ . This group has an outer automorphism, and so has two different actions on sets of size $6$ . Take $\Omega $ to be the Cartesian product of these two sets. The invariant partitions for G are E and U together with the rows R and columns C of the square. Then $\{E,R,C,U\}$ is a poset block structure. Both R and C are join-indecomposable, and $R^-=C^-=E$ . Thus M is a $2$ -element antichain $\{r,c\}$ , and $G(r)$ is the stabiliser of a row acting on the points of the row, which is the group $\mathrm {PGL}(2,5)$ , and similarly $G(c)$ . However, $S_6$ is clearly not embeddable in $\mathrm {PGL}(2,5)\times \mathrm {PGL}(2,5)$ .

So we use a more complicated construction. Given $\Pi $ and $\Pi ^-$ as above, where $\Pi $ corresponds to $m\in M$ , let $\mathcal {G}(m)$ be the set of partitions $\Phi \in \Lambda $ satisfying $\Phi \wedge \Pi =\Pi ^-$ . For $\Phi \in \mathcal {G}(m)$ , let $G_\Phi (m)$ be the group induced on the set of parts of $\Phi $ contained in a given part of $\Phi \vee \Pi $ .

Proof. The first part is immediate from the distributive law: if $\Phi _1,\Phi _2\in \mathcal {G}(m)$ , then

$$\begin{align*}(\Phi_1\vee\Phi_2)\wedge\Pi=(\Phi_1\wedge\Pi)\vee(\Phi_2\wedge\Pi) =\Pi^-\vee\Pi^-=\Pi^-.\end{align*}$$

For the second part, we use the fact that, for a given point $\alpha \in \Omega $ , there is a natural correspondence between partitions and certain subgroups of G containing $G_\alpha $ , where the partition $\Pi $ corresponds to the setwise stabiliser of the part of $\Pi $ containing $\alpha $ ; meet and join correspond to intersection and product of subgroups. Let $H_1$ , $H_2$ , P, $P^-$ be the subgroups corresponding to $\Phi _1$ , $\Phi _2$ , $\Pi $ , $\Pi ^-$ . Then the definition of $\mathcal {G}(m)$ shows that $H_i\cap P=P^-$ for $i=1,2$ , while the partitions $\Phi _i\vee \Pi $ correspond to the subgroups $H_iP$ . The actions we are interested in are thus $H_iP$ on the cosets of $H_i$ . We have

$$\begin{align*}|H_iP:H_i|=|P:H_i\cap P|=|P:P^-|\end{align*}$$

for $i=1,2$ ; so coset representatives of $P^-$ in P are also coset representatives for $H_i$ in $H_iP$ . Thus we have a natural correspondence between these sets. Since $H_1\le H_2$ , we have $H_1P\le H_2P$ , and the result holds.

Hence if $\Psi $ is the (unique) maximal element of $\mathcal {G}(m)$ , then the group $G_\Psi (m)$ , which we will denote by $G^*(m)$ , embeds all the groups $G_\Phi (m)$ for $\Phi \in \mathcal {G}(m)$ .

Now we can state the embedding theorem.

We remark that this theorem generalises the theorem of Krasner and Kaloujnine. If $\Pi $ is a nontrivial G-invariant partition, then $\{E,\Pi ,U\}$ is a poset block structure; the corresponding poset is $M=\{m_1,m_2\}$ , with $m_1$ and $m_2$ corresponding to the partitions $\Pi $ and U; this $G^*(m_1)$ is the group induced by the stabiliser of a part of $\Pi $ on its points, and $G^*(m_2)$ the group induced by G on the parts of $\Pi $ , as required.

The proof uses properties of distributive lattices: we deal with some of these first. Since these lemmas are not specifically about lattices of partitions, we depart from our usual convention and use lower-case italic letters for elements of a lattice, and $0$ and $1$ for the least and greatest elements respectively.

Lemma 4.22. Let L be a distributive lattice. If $a,x,y\in L$ satisfy

$$\begin{align*}a\wedge x=a\wedge y\text{ and }a\vee x=a\vee y, \end{align*}$$

then $x=y$ .

Proof. Suppose first that $x\le y$ . Then

$$ \begin{align*} y &= y\vee(a\wedge x) \\ &= (y\vee a)\wedge(y\vee x) \\ &= (x\vee a)\wedge(x\vee y) \\ &= x\vee(a\wedge y) \\ &= x\vee(a\wedge x) \\ &= x. \end{align*} $$

Now let x and y be arbitrary, and put $z=x\wedge y$ . Then $z\le x$ and

$$ \begin{align*} a\wedge z &= (a\wedge x)\wedge(a\wedge y)=a\wedge x\\ a\vee z &= (a\vee x)\wedge (a\vee y)=a\vee x. \end{align*} $$

By the first part, $z=x$ . Similarly $z=y$ , so $x=y$ .

Proof. Let $z=\{p\}$ . Let $\mathop {\mathrm {JI}}(L)$ be the set of join-indecomposables in L. Since lattices are generated by their join-indecomposable elements, it suffices to construct an order-isomorphism F from $\mathop {\mathrm {JI}}(L)\setminus z$ to $\mathop {\mathrm {JI}}([z,1])$ . The map F is defined by

$$\begin{align*}F(a)=a\vee z\end{align*}$$

for $a\in \mathop {\mathrm {JI}}(L)\setminus \{z\}$ . We have to show that it is a bijection and preserves order. First we show that its image is contained in $\mathop {\mathrm {JI}}([z,1])$ .

Take $a\in \mathop {\mathrm {JI}}(L)$ , $a\ne z$ . If $z\le a$ , then $a\vee z=a$ and this is join-indecomposable in $[z,1]$ . Suppose that $z\not \le a$ . If $a\vee z$ is not JI in $[z,1]$ , then there exist $b,c\in [z,1]$ with $b,c\ne a\vee z$ and $b\vee c=a\vee z$ . Then

$$\begin{align*}(a\wedge b)\vee(a\wedge c)=a\wedge(b\vee c)=a\wedge(a\vee z)=a.\end{align*}$$

Since a is join-indecomposable, we have, without loss of generality, $a\wedge b=a$ , so $a\le b$ . Since we also have $z\le b$ , it follows that $a\vee z \le b$ , and so $a\vee z = b$ , a contradiction.

We show that the map is onto. Let $a\in \mathop {\mathrm {JI}}([z,1])$ . If $a\in \mathop {\mathrm {JI}}(L)$ then $a=F(a)$ ; so suppose not. Then $a=b\vee c$ for some $b,c\in L$ . Then

$$\begin{align*}z=a\wedge z=(b\vee c)\wedge z=(b\wedge z)\vee(c\wedge z),\end{align*}$$

so at least one of b and c (but not both) is in $[z,1]$ , say $b\in [z,1]$ . Then $a=b\vee (c\vee z)$ . Since $a\in \mathop {\mathrm {JI}}([z,1])$ and $a\ne b$ , we must have $c\vee z=a$ . We claim that c is join-indecomposable. For if $c=d\vee e$ , then

$$\begin{align*}a=c\vee z=(d\vee z)\vee(e\vee z).\end{align*}$$

If $d\vee z=a=c\vee z$ , then $c=d$ (since $d\wedge z=0=c\wedge z$ ), a contradiction. The other case leads to a similar contradiction.

Next we show that F is one-to-one. Suppose that $F(a_1)=F(a_2)$ . If $a_1,a_2\in [z,1]$ , then $a_1=a_2$ . If $a_1,a_2\notin [z,1]$ , then $a_1\vee z=F(a_1)=F(a_2)=a_2\vee z$ ; also $a_1\wedge z=0=a_2\wedge z$ . By Lemma 4.22, $a_1=a_2$ . So suppose that $a_1\in [z,1]$ , $a_2\notin [z,1]$ . Then $a_1=F(a_1)=F(a_2)=a_2\vee z$ , contradicting the fact that $a_1$ is join-indecomposable.

Finally we show that F is order-preserving. Suppose that $a_1\le a_2$ . If $a_1,a_2\in [z,1]$ , then $F(a_1)=a_1\le a_2=F(a_2)$ . If $a_1,a_2\notin [z,1]$ , then $F(a_1)=a_1\vee z\le a_2\vee z=F(a_2)$ . We cannot have $a_1\in [z,1]$ and $a_2\notin [z,1]$ , since then $z\le a_1\le a_2$ but $z\not \le a_2$ . Finally suppose that $a_1\notin [z,1]$ but $a_2\in [z,1]$ , so that $z\le a_2$ and $a_1\le a_2$ , then $F(a_1)=a_1\vee z\le a_2=F(a_2)$ .

Now we turn to the proof of Theorem 4.21. The proof is by induction on the number of elements in M. We take $\Pi _0$ to be a minimal non-E partition, corresponding to a minimal element $p\in M$ . We decorate things computed in the interval $[\Pi _0,U]$ with bars; for example, $\bar G^*(q)$ corresponds to the group associated in this lattice with the element $q\ne p$ (which is not in general the same as $G^*(q)$ ). Thus $\bar G$ is the group induced by G on the set of parts of $\Pi _0$ , which is a PB group with associated poset $M\setminus \{p\}$ ; our induction hypothesis will imply that the group $\bar G$ is embedded in the generalised wreath product of the groups $\bar G^*(q)$ for $q\in M\setminus \{p\}$ .

Let $\Pi $ be a join-indecomposable partition in $[\Pi _0,U]$ , corresponding to the element $q\in M\setminus \{p\}$ . As we saw in the proof of Lemma 4.23, there are two possibilities:

• ○ Case 1: $\Pi $ is join-indecomposable in the lattice L of downsets of M. Then $\Pi ^-$ is above $\Pi _0$ , and so the group $\bar G^*(q)$ is the same as $G^*(q)$ .

• ○ Case 2: $\Pi =\Pi _0\vee \Psi $ , where $\Psi $ is join-indecomposable in L and $\Pi _0$ is not below $\Psi $ . Consider the set $\bar {\mathcal {G}}(q)$ , where the bar denotes that it is computed in the lattice $[\Pi _0,U]$ . A partition $\Phi $ belongs to this set if it is above $\Pi _0$ and satisfies $\Phi \wedge \Pi =\bar \Pi ^{-}$ , where again the bar denotes the unique maximal element below $\Pi $ in $[\Pi _0,U]$ . An easy exercise shows that $\bar \Pi ^{-}\wedge \Psi =\Psi ^-$ ; hence

  $$\begin{align*}\Phi\wedge\Psi=\Psi^-,\end{align*}$$
  and so $\Phi $ belongs to $\mathcal {G}(q)$ . In other words, we have shown that
  $$\begin{align*}\bar{\mathcal{G}}(q)\subseteq\mathcal{G}(q).\end{align*}$$

  By Lemma 4.20, $\bar G^*(q)$ is canonically embedded in $G^*(q)$ .