# A survey on association schemes on triples

Jose Maria P. Balmaceda  Institute of Mathematics, University of the
Philippines Diliman, 1101 Quezon City, Philippines<EMAIL_ADDRESS>and
Dom Vito A. Briones  Institute of Mathematics, University of the Philippines
Diliman, 1101 Quezon City, Philippines<EMAIL_ADDRESS>
###### Abstract.

Association schemes on triples (ASTs) are ternary analogues of classical
association schemes, whose relations and adjacency algebras are ternary
instead of binary. We provide a survey of the current progress in the study of
ASTs, highlighting open questions, suggesting research directions, and
producing some related results. We review properties of the ternary adjacency
algebras of ASTs, ASTs whose relations are invariant under some group action,
and ASTs obtained from 2-designs and two-graphs. We also provide a notion of
fusion and fission ASTs, using the AST obtained from the affine special linear
group $ASL(2,q)$ as an example.

###### Key words and phrases:

algebraic combinatorics, ternary algebra, association scheme on triples
MSC Classification: 05E30

##  1\. Introduction

Classical association schemes originated from Bose and Shimamoto as partitions
of a Cartesian product $\Omega\times\Omega$ with certain symmetry properties
[5]. These may be regarded as colorings of the edges of complete graphs
satisfying desirable regularity conditions. A special case is the family of
two-class association schemes which consists of colorings with two colors that
yield the family of strongly regular graphs. The symmetry conditions that
classical association schemes satisfy are sufficiently flexible as to
accommodate various mathematical structures, yet adequately rigid as to endow
classical association schemes with many desirable algebraic and combinatorial
properties [3]. For instance, the algebras generated by the adjacency matrices
of classical association schemes are semisimple and, when commutative, satisfy
duality properties that allow for the computation of possible parameter values
of various families of graphs.

In [11], Mesner and Bhattacharya defined a ternary analogue for classical
association schemes called association schemes on triples (ASTs). Instead of
partitions of the Cartesian product $\Omega\times\Omega$, an AST is a
partition of the Cartesian triple product $\Omega\times\Omega\times\Omega$
satisfying analogous regularity conditions. In particular, the adjacency
hypermatrices of ASTs form a ternary algebra under an extension of the usual
binary matrix product to a ternary cubic hypermatrix product.

In this survey, we share current progress in the study of ASTs, highlighting
open questions, suggesting research directions, and producing some related
results. In particular, we consider properties of the ternary adjacency
algebras of ASTs, ASTs whose relations are invariant under some group action,
such as symmetric ASTs, ASTs from two-transitive groups, and circulant ASTs,
and the ASTs obtained from 2-designs and two-graphs. Lastly, we provide a
notion of fusion and fission ASTs, using the AST obtained from the affine
special linear group $ASL(2,q)$ as a primary example. The parameters of this
AST are also obtained, thereby extending the work done in [2]. The paper is
structured as follows. We define ASTs in Section 2 and consider its algebraic
structure as a ternary algebra in Section 3. Through Section 4, we examine the
relationships between group actions and ASTs. Proceeding, we explore the
relationships between ASTs and 2-designs and two-graphs in Section 5. Finally,
we define fusion and fission ASTs in Section 6, providing some examples and
focusing particularly upon the AST obtained from $ASL(2,q)$.

##  2\. Assocation schemes on triples

This section is based mostly on [11] and [13]. We define an association scheme
on triples (AST) as a partition of a triple cartesian product satisfying
certain symmetry conditions and then view ASTs as ternary algebras through
their adjacency hypermatrices.

######  Definition 2.1.

Let $\Omega$ be a finite set with at least 3 elements. An association scheme
on triples (AST) on $\Omega$ is a partition $X=\\{R_{i}\\}_{i=0}^{m}$ of
$\Omega\times\Omega\times\Omega$ with $m\geq 4$ such that the following hold.

  1. (1)

For each $i\in\\{0,\ldots,m\\}$, there exists an integer $n_{i}^{(3)}$ such
that for each pair of distinct $x,y\in\Omega$, the number of $z\in\Omega$ with
$(x,y,z)\in R_{i}$ is $n_{i}^{(3)}$.

  2. (2)

(Principal Regularity Condition.) For any $i,j,k,l\in\\{0,\ldots,m\\}$, there
exists a constant $p_{ijk}^{l}$ such that for any $(x,y,z)\in R_{l}$, the
number of $w$ such that $(w,y,z)\in R_{i}$, $(x,w,z)\in R_{j}$, and
$(x,y,w)\in R_{k}$ is $p_{ijk}^{l}$.

  3. (3)

For any $i\in\\{0,\ldots,m\\}$ and any $\sigma\in S_{3}$, there exists a
$j\in\\{0,\ldots,m\\}$ such that

$R_{j}=\\{(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}):(x_{1},x_{2},x_{3})\in
R_{i}\\}.$

  4. (4)

The first four relations are $R_{0}=\\{(x,x,x):x\in\Omega\\}$,
$R_{1}=\\{(x,y,y):x,y\in\Omega,x\neq y\\}$,
$R_{2}=\\{(y,x,y):x,y\in\Omega,x\neq y\\}$, and
$R_{3}=\\{(y,y,x):x,y\in\Omega,x\neq y\\}$.

The following example is the AST on $\Omega=\\{1,2,3\\}$ with five relations.

######  Example 2.2.

Let $\Omega=\\{1,2,3\\}$ and $X=\\{R_{i}\\}_{i=0}^{4}$ be the partition of
$\Omega\times\Omega\times\Omega$ given by the following ternary relations.

$\displaystyle{R_{0}}$ $\displaystyle=\\{(1,1,1),(2,2,2),(3,3,3)\\},$
$\displaystyle{R_{1}}$
$\displaystyle=\\{(1,2,2),(1,3,3),(2,1,1),(2,3,3),(3,1,1),(3,2,2)\\},$
$\displaystyle{R_{2}}$
$\displaystyle=\\{(2,1,2),(3,1,3),(1,2,1),(3,2,3),(1,3,1),(2,3,2)\\},$
$\displaystyle{R_{3}}$
$\displaystyle=\\{(2,2,1),(3,3,1),(1,1,2),(3,3,2),(1,1,3),(2,2,3)\\},$
$\displaystyle R_{4}$
$\displaystyle=\\{(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)\\}.$

Then $X$ is an AST on $\Omega$. Since $3$ is the only $w\in\Omega$ such that
$(1,2,w)\in R_{4}$, we have $n_{4}^{(3)}=1$. Further, $(1,2,3)\in R_{4}$ and
there is no $w$ such that $(w,2,3)$, $(1,w,3)$, and $(1,2,w)$ are all in
$R_{4}$. Thus, we have $p_{444}^{4}=0$.

The integer $n_{i}^{(3)}$ is the third valency of $R_{i}$, analogous to the
valency of a classical association scheme. Conditions 1 and 3 imply for each
$i$ the existence of constants $n_{i}^{(1)}=|\\{z\in\Omega:(z,x,y)\in
R_{i}\\}|$ and $n_{i}^{(2)}=|\\{z\in\Omega:(x,z,y)\in R_{i}\\}|$ independent
of any pair of distinct $x,y\in\Omega$. Accordingly, $n_{i}^{(1)}$ is called
the first valency of $R_{i}$ and $n_{i}^{(2)}$ is called the second valency of
$R_{i}$. The relations $R_{0},R_{1},R_{2}$ and $R_{3}$ are called the trivial
relations and the other relations are the nontrivial relations. Further, the
numbers $p_{ijk}^{l}$ are called the intersection numbers.

An AST can be viewed in terms of hypermatrices that generate a ternary algebra
whose structure constants are the $p_{ijk}^{l}$, mirroring the situation
between classical association schemes and their adjacency algebras.

Let $X=\\{R_{i}\\}_{i=0}^{m}$ be an AST on a set $\Omega$ of size $\nu$. We
associate with each $R_{i}\in X$ the $\nu\times\nu\times\nu$ hypermatrix
$A_{i}$ whose entries are indexed by $\Omega$. This $A_{i}$ is given by

$(A_{i})_{xyz}=\begin{cases}1,&\text{if }(x,y,z)\in R_{i},\\\
0,&\text{otherwise}.\end{cases}$

The $\mathbb{C}$-vector space generated by the adjacency hypermatrices forms a
ternary algebra under the ternary operation $ABC\mapsto D$, where $D$ is the
$\nu\times\nu\times\nu$ hypermatrix given by

$D_{xyz}=\sum_{w\in\Omega}A_{wyz}B_{xwz}C_{xyw}.$

The structure constants of this ternary algebra are given by the intersection
numbers $p_{ijk}^{l}$ of $X$ [11]; that is,
$A_{i}A_{j}A_{k}=\sum_{l=0}^{m}p_{ijk}^{l}A_{l}$. For instance, if we consider
the AST in Example 2.2, then we obtain
$A_{4}A_{4}A_{4}=\sum_{l\in\Omega}p_{444}^{l}A_{l}=0$.

##  3\. Algebraic structure of ASTs

Currently, little is known about the structure of the ternary algebra obtained
from the adjacency matrices of an AST. However, the situation is partially
simplified by considering the subalgebra generated by the adjacency
hypermatrices of the nontrivial relations. Indeed, Corollary 2.8 of [11] says
that the adjacency matrices $\\{A_{i}\\}_{i=4}^{m}$ of the nontrivial
relations generate a subalgebra of the ternary algebra generated by all the
adjacency matrices $\\{A_{i}\\}_{i=0}^{m}$. As other theorems and remarks in
[11] provide values and restrictions for $p_{ijk}^{l}$ when
$\\{i,j,k,l\\}\cap\\{0,1,2,3\\}\neq\varnothing$, [11] suggests that the most
interesting intersection numbers are those that arise from the above
subalgebra.

One open problem is to determine whether or not the desirable algebraic
properties of classical association schemes hold for ASTs, such as the
semisimplicity and duality properties of their adjacency algebras. In [10],
Lister develops a structure theory of associative ternary algebras, with
analogues of ideals, modules, identities, inverses, and decompositions. In
particular, he provides ternary analogues for fields and semisimplicity. The
simplest case where we may apply [10] occurs when the given AST has only one
nontrivial relation.

######  Example 3.1.

Let $X=\\{R_{i}\\}_{i=0}^{4}$ be an AST with only one nontrivial relation
$R_{4}$. In this case, the ternary subalgebra generated by $A_{4}$ is both
associative and commutative. In fact, with $A_{4}A_{4}A_{4}=p_{444}^{4}A_{4}$,
we see that $\frac{1}{p_{444}}A_{4}$ and $A_{4}$ is a ternary identity of the
subalgebra provided that $p_{444}^{4}\neq 0$. Moreover, with $c\neq 0$,
$cA_{4}$ has an inverse $\frac{1}{cp_{444}^{4}}A_{4}$. Therefore, when there
is only one nontrivial relation $R_{4}$, the subalgebra generated by $A_{4}$
is a ternary field.

Beyond this, we currently have no examples of ASTs whose subalgebras generated
by the adjacency hypermatrices of the nontrivial relations are associative. As
such, we have yet to acquire nontrivial examples on which to apply the
structure theory of [10]. A possible means to circumvent this may be taken
from [12], where a different notion of identities and inverses, called
identity pairs and inverse pairs, are defined. A means of computation of such
pairs are also given, with applications to ASTs obtained from 2-designs and
some two-transitive groups. Although the authors did not provide a direct
application of such pairs to ASTs, they found that computing for inverse pairs
is equivalent to solving certain equations with the structure constants of the
ASTs, and that there are plenty of inverse pairs from the subalgebras of ASTs.
More recently, Gnang derived explicit necessary and sufficient conditions for
the existence of inverse pairs and used these to formulate and prove a ternary
analogue for the usual rank-nullity theorem [9].

If we instead focus on commutativity, there are nontrivial examples of
commutative association schemes that will be given in the later sections. In
particular, the ASTs from 2-designs, and the ASTs from the projective linear
groups and the sporadic two-transitive groups are commutative. However, the
structural properties of the subalgebras afforded by their commutativity
remain unknown. In the classical association scheme case, the semisimplicity
and duality properties may be obtained from applications of the spectral
decomposition theorem on the mutually commuting adjacency matrices [3].
Although they have yet to be applied to study the structure of the subalgebras
of ASTs, there exist analogues of the fundamental theorem of linear algebra,
the spectral theorem, and the Gram-Schmidt orthogonalization process for cubic
hypermatrices [8, 7].

##  4\. ASTs from group actions

In this section we consider ASTs whose relations are invariant under the
action of some group.

###  4.1. ASTs from two-transitive groups

Analogous to Schurian association schemes, which are classical association
schemes obtained from transitive group actions [3], an AST may arise from the
action of a two-transitive group. Indeed, given a two-transitive group $G$
acting on a set $\Omega$, the orbits of the induced action on
$\Omega\times\Omega\times\Omega$ is an AST [11]. Further, the orbits of a two-
point stabilizer are in bijection with the nontrivial relations of the AST and
the sizes of these orbits give the third valencies [11, 2].

In [11], the sizes of the ASTs from the affine group $AGL(1,n)$, the
projective group $PSL(2,n)$, the Suzuki group $Sz(2^{2k+1})$, and the Higman-
Sims group $HS$ were acquired. Some intersection numbers from the ASTs
obtained from $AGL(1,n)$ and $PSL(2,n)$ were also obtained. These were
extended in [2], where the sizes and third valencies of the ASTs obtained from
$S_{n}$ and $A_{n}$, $PGU(3,q)$, $PSU(3,q)$, and $Sp(2k,2)$, $Sz(2^{2k+1})$
and $Ree(3^{2k+1})$, some subgroups of $A\Gamma L(k,n)$, some subgroups of
$P\Gamma L(k,n)$, and the sporadic two-transitive groups, were obtained. The
intersection numbers of the ASTs from these subgroups of $P\Gamma L(k,n)$ and
$A\Gamma L(k,n)$, and the sporadic two-transitive groups were also determined.
In particular, the ASTs obtained from the projective linear groups and the
sporadic two-transitive groups were found to be commutative.

As of yet, the intersection numbers of the ASTs from the projective unitary
groups, the symplectic groups, the Suzuki groups, and the Ree groups remain
undetermined. The sizes, third valencies, and intersection numbers of the two-
transitive subgroups of $A\Gamma L(k,n)$ not of the form $AGL(k,n)\rtimes H$
(where $H\leq Gal(GF(n))$) also remain undetermined.

###  4.2. Symmetric and Circulant ASTs

A ternary relation $R\subseteq\Omega\times\Omega\times\Omega$ is called
symmetric if it is invariant under coordinate permutation; that is, for any
$\sigma\in S_{3}$, we have

$R=\\{(x_{\sigma(1)},x_{\sigma(2)},x_{(\sigma(3))}):(x,y,z)\in R\\}.$

An AST is called symmetric if all its nontrivial relations are symmetric [11].
In [11], some parameters of such ASTs are computed. Through these
computations, the authors found that a weak associative law holds for
symmetric ASTs:

$(A_{i}A_{i}A_{j})A_{i}A_{i}=A_{i}(A_{i}A_{j}A_{i})A_{i}=A_{i}A_{i}(A_{j}A_{i}A_{i}).$

Some examples of symmetric ASTs are the ASTs with only one nontrivial
relation, and ASTs obtained from 2-designs and two-graphs [11], as will be
discussed in the next section.

Motivated by symmetric ASTs, another family of ASTs called circulant ASTs was
defined in [13]. The nontrivial relations of these ASTs are called circulant,
being invariant under the action of a common transitive cyclic subgroup of
$S_{n}$. It turns out that such ASTs correspond to partitions of a certain
subset of $\Omega^{[2]}=\\{(x,y):x\neq y\\}\subseteq\Omega\times\Omega$ called
AST-regular partitions. In particular, enumerating the AST-regular partitions
yields all circulant ASTs over $\Omega$. This suggests looking for AST-regular
partitions from known families of circulant ASTs, particularly the ASTs
obtained from $PSL(2,q)$ for $q$ even and $AGL(1,q)$ for $q$ prime.

The authors of [13] also defined the notion of thinness for ternary relations,
wherein a circulant ternary relation $R$ is said to be $ab$-thin provided that
the mapping $\sigma_{ab}$ from $R$ to $\Omega\times\Omega$ given by
$\sigma_{ab}(x_{1},x_{2},x_{3})=(x_{a},x_{b})$ is injective with image
$\Omega^{[2]}$. By reasoning with thin relations, they showed that any
nontrivial relation of a circulant AST is a disjoint union of thin circulant
ternary relations. In light of this, [13] suggests finding circulant ASTs
where every nontrivial circulant relation is thin.

The authors of [13] also suggested finding further examples of circulant ASTs.
We give a new example below, obtained through GAP 4.11.1, which is the AST
obtained from the action of the permutation representation of $PSL(2,11)$ of
degree 11. It is a commutative AST whose parameters are given in [2].

######  Example 4.1.

Let $G$ be the subgroup of $S_{11}$ produced by the following generators:

$(1,2,4,9,5,7,3,11,10,6,8),\;(1,2,5,8,9)(4,10,7,6,11),\text{ and
}(1,2,5,8,10,11)(3,6,7)(4,9).$

Then $G$ is permutation isomorphic to the degree 11 representation of
$PSL(2,11)$ as described in [6]. Further, the AST obtained from $G$ is
circulant, as each of its nontrivial relations is invariant under the action
of the subgroup of $S_{11}$ generated by the length 11 cycle
$(1,2,3,10,6,11,8,4,5,9,7)$.

###  4.3. ASTs with relations invariant under some group

Motivated by the classification of classical association schemes over small
vertices and the computational simplifications that arise by considering ASTs
whose relations are invariant under some group action, [1] provided an
algorithm for generating (up to isomorphism) all ASTs over a given number of
vertices whose nontrivial relations are invariant under some predetermined
group action. In particular, appropriate choices of group actions can
enumerate the symmetric ASTs, circulant ASTs, and even all ASTs over a given
number of vertices, provided sufficient computational resources are available.

The authors found that there is a unique AST over three vertices, a unique
symmetric ASTs over four or five vertices, a unique AST over four vertices
with two nontrivial relations, and a unique nontrivial circulant AST over five
vertices. Since the current version of the algorithm is computationally
expensive, it would benefit from improvements either in the algorithm itself
or in its implementation.

##  5\. ASTs from 2-designs and two-graphs

Although there are no constructions at this time of ASTs that serve as
analogues of the Hamming scheme or the Johnson scheme from classical
association schemes, there are relationships between ASTs and other
combinatorial objects such as two-graphs and 2-designs [11].

###  5.1. 2-designs

A 2-design $(\Omega,B)$ with parameters $b,v,k,\lambda$ is a family $B$ of
$k$-subsets of a set $\Omega$ (called blocks) such that any $2$-subset of
$\Omega$ lies in exactly $\lambda$ blocks, and where $|\Omega|=v$, and
$|B|=b$. In [11], it was found that any symmetric nontrivial relation $R_{i}$
of an AST yields a family of 2-designs. Indeed, the family $B$ of all
3-subsets $\\{x,y,z\\}$ such that $(x,y,z)\in R_{i}$ is a 2-design. Moreover,
a 2-design $B$ with $\lambda=1$ yields an AST, by letting the nontrivial
relations be

$R_{4}=\\{(x,y,z):x,y,z\;\text{distinct, }\\{x,y,z\\}\;\text{lies in some
block of }B\\}$

and

$R_{5}=\\{(x,y,z):x,y,z\;\text{distinct, }\\{x,y,z\\}\;\text{does not lie in
any block of }B\\}.$

The authors then proved that this subalgebra generated by the adjacency
matrices $A_{4}$ and $A_{5}$ is commutative but not associative. It is also
not minimal, for $A_{4}$ generates a proper subalgebra not containing $A_{5}$.
As a converse, an AST with two nontrivial relations that are both symmetric
and whose parameters satisfy certain conditions can be recovered from a
2-design through the above construction [11].

The construction above is for designs with $\lambda=1$. However, it is
generally not possible to proceed in a similar manner for larger values of
$\lambda$, although the construction holds for certain choices of
$\lambda$-designs [11]. It remains undetermined which $\lambda$ and
$\lambda$-designs permit such a construction of an AST.

###  5.2. Two-graphs

A collection $\Delta$ of $3$-subsets of a set $\Omega$ such that every
$4$-subset of $\Omega$ contains an even number of members of $\Delta$ is
called a two-graph $(\Omega,\Delta)$. A two-graph is regular if each pair of
elements of $\Omega$ is contained in the same number of triples of $\Delta$.
In [11], the authors showed that an AST with two nontrivial relations $R_{4}$
and $R_{5}$ that are both symmetric and whose parameters satisfy

$p_{445}^{4}=p_{454}^{4}=p_{544}^{4}=p_{555}^{4}=p_{444}^{5}=p_{455}^{5}=p_{545}^{5}=p_{554}^{4}=0$

is equivalent to a regular two-graph.

##  6\. Fusion and fission ASTs

Motivated by classical fusion and fission association schemes [4], we define
fusion and fission ASTs, providing a relationship between fusion and fission
ASTs and a number of examples. In particular, we give the AST obtained from
$ASL(2,q)$ as an instance of a fission scheme of the AST obtained from the
affine general linear group $AGL(2,q)$. Further, we obtain the parameters of
the AST from $ASL(2,q)$, extending the results of [2] to another infinite
family of two-transitive groups. We begin with the definition of fusion and
fission ASTs, ASTs that can be obtained from combining or splitting the
relations of another AST.

######  Definition 6.1.

Let $\Omega$ be a nonempty set and let $X=\\{R_{i}\\}_{i=0}^{m}$ and
$\tilde{X}=\\{\tilde{R}_{\alpha}\\}_{\alpha=0}^{n}$ be ASTs on $\Omega$. If
for each $i\in\\{0,\ldots,m\\}$ there exists an $\alpha\in\\{0,\ldots,n\\}$
such that $R_{i}\subseteq\tilde{R}_{\alpha}$, then we say that $\tilde{X}$ is
a fusion AST of $X$ and $X$ is a fission AST of $\tilde{X}$.

In other words, the relations of a fusion AST $\tilde{X}$ are unions of the
relations of its fission AST $X$. The following theorem provides a
relationship between the intersection numbers of a fission AST and a fusion
AST.

######  Theorem 6.2.

Let $\Omega$ be a nonempty set, $X=\\{R_{i}\\}_{i=0}^{m}$ be an AST on
$\Omega$, and $\tilde{X}=\\{\tilde{R}_{\alpha}\\}_{\alpha=0}^{n}$ be a fusion
AST of $X$. For each $\alpha\in\\{0,\ldots,n\\}$, let
$\Lambda_{\alpha}=\\{i\in\\{0,\ldots,m\\}:R_{i}\subseteq\tilde{R}_{\alpha}\\}$.
If $\tilde{p}_{\alpha\beta\gamma}^{\delta}$ is the intersection number
corresponding to $\tilde{R}_{\alpha}$, $\tilde{R}_{\beta}$,
$\tilde{R}_{\gamma}$, and $\tilde{R}_{\delta}$, then

$\tilde{p}_{\alpha\beta\gamma}^{\delta}=\sum_{i\in\Lambda_{\alpha}}\sum_{j\in\Lambda_{\beta}}\sum_{k\in\Lambda_{\gamma}}p_{ijk}^{l}$

for any $l\in\Lambda_{\delta}$. Furthermore, the third valency of a nontrivial
relation $\tilde{R}_{\epsilon}$ of $\tilde{X}$ is
$\tilde{n}_{\epsilon}^{(3)}=\sum_{i\in\Lambda_{\epsilon}}n_{i}^{(3)}$.

###### Proof.

Fix any $(x,y,z)\in R_{l}\subseteq\tilde{R}_{\delta}$. The intersection number
$\tilde{p}_{\alpha\beta\gamma}^{\delta}$ is equal to the number of $w$ such
that $(w,y,z)\in\tilde{R}_{\alpha}$, $(x,w,z)\in\tilde{R}_{\beta}$, and
$(x,y,w)\in\tilde{R}_{\gamma}$. However, these conditions on $w$ are
equivalent to $(w,y,z)\in R_{i}$ for some $i\in\Lambda_{\alpha}$, $(x,w,z)\in
R_{j}$ for some $j\in\Lambda_{\beta}$, and $(x,y,w)\in R_{k}$ for some
$k\in\Lambda_{\gamma}$. This yields

$\tilde{p}_{\alpha\beta\gamma}^{\delta}=\sum_{i\in\Lambda_{\alpha}}\sum_{j\in\Lambda_{\beta}}\sum_{k\in\Lambda_{\gamma}}p_{ijk}^{l}.$

Finally, given a nontrivial relation $\tilde{R}_{\epsilon}$ of $\tilde{X}$ and
a member $(x,y,z)$ of $\tilde{R}_{\epsilon}$, the third valency
$\tilde{n}_{\epsilon}^{(3)}$ is equal to the number of $w$ such that
$(x,y,w)\in\tilde{R}_{\epsilon}$. This condition on $w$ is equivalent to
$(x,y,w)$ being in $R_{i}$ for some $i\in\Lambda_{\epsilon}$, yielding
$\tilde{n}_{\epsilon}^{(3)}=\sum_{i\in\Lambda_{\epsilon}}n_{i}^{(3)}$. ∎

Theorem 6.2 can be expressed in terms of the adjacency hypermatrices of $X$
and $\tilde{X}$. If for each $\alpha\in\\{0,\ldots,n\\}$ we let
$\tilde{A}_{\alpha}$ denote the adjacency hypermatrix corresponding to
$\tilde{R}_{\alpha}\in\tilde{X}$, then
$\tilde{A}_{\alpha}=\sum_{i\in\Lambda_{\alpha}}{A_{i}}$. Hence, we have the
following equality for any $\alpha,\beta,\gamma\in\\{0,\ldots,n\\}$.

$\tilde{A}_{\alpha}\tilde{A}_{\beta}\tilde{A}_{\gamma}=\sum_{\delta=0}^{n}\tilde{p}_{\alpha\beta\gamma}^{\delta}\tilde{A}_{\delta}=\sum_{\delta=0}^{n}\sum_{l\in\Lambda_{\delta}}\tilde{p}_{\alpha\beta\gamma}^{\delta}A_{l}=\sum_{\delta=0}^{n}\sum_{l\in\Lambda_{\delta}}\left(\sum_{i\in\Lambda_{\alpha}}\sum_{j\in\Lambda_{\beta}}\sum_{k\in\Lambda_{\gamma}}p_{ijk}^{l}\right)A_{l}.$

We now give some examples of fusion and fission ASTs. First, the AST over a
set $\Omega$ with only one nontrivial relation is a fusion scheme of any AST
over $\Omega$.

######  Example 6.3.

Let $X=\\{R_{i}\\}_{i=0}^{m}$ be any AST on a set $\Omega$. Let
$\tilde{X}=\\{R_{0},R_{1},R_{2},R_{3},\cup_{i=4}^{m}R_{i}\\}$ be the given
partition of $\Omega\times\Omega\times\Omega$. Then $\tilde{X}$ is a fusion
AST of $X$ equal to the only AST over $\Omega$ with only one nontrivial
relation.

######  Example 6.4.

The following example from two-graphs is obtained from Remark 5.6 of [11]. If
an AST $X=\\{R_{\alpha}\\}_{\alpha=0}^{n}$ that has more than two nontrivial
relations contains a subset $J\subseteq\\{4,5,\ldots,m\\}$ of indices such
that ${R}_{i}$ for each $i\in J$ is symmetric, define

$\Delta=\\{\\{a,b,c\\}:(a,b,c)\in R_{i}\text{ for some }i\in J\\}.$

If for any quadruple $i,j,k,l$ from $\Omega$ such that an odd number of these
lie in $J$ we have ${p}_{ijk}^{l}=0$, then $\Delta$ is a regular two-graph.
This two-graph then produces an AST $\tilde{X}$ with two-nontrivial relations,
each of which is a union of the nontrivial relations of $X$; that is,
$\tilde{X}$ is a fusion AST of $X$.

The next example shows that fission ASTs occur naturally from two-transitive
subgroups of two-transitive groups.

######  Example 6.5.

Let $G$ be a two-transitive group acting on $\Omega$ and $H$ be a two-
transitive subgroup of $G$. Let $\tilde{X}$ be the AST obtained from the
action of $G$ on $\Omega$ and $X$ be the AST obtained from the action of $H$
on $\Omega$. Then $X$ is a fission scheme of $\tilde{X}$. Indeed, the
relations of $X$ are the orbits of $H$ on $\Omega\times\Omega\times\Omega$
while the relations of $\tilde{X}$ are the orbits of $G$ on
$\Omega\times\Omega\times\Omega$. Since $H\leq G$, the orbits of $G$ on
$\Omega\times\Omega\times\Omega$ are unions of orbits of $H$. Examples of such
fission ASTs occuring from two-transitive subgroups of two-transitive groups,
along with some of their parameters, are in [2].

For the final illustration, we compute the parameters of the AST obtained from
the affine special linear group $ASL(2,q)$ where $q$ is a prime power,
extending the work done in [2]. Since $ASL(2,q)$ is a two-transitive subgroup
of $AGL(2,q)$, the following is also an example of a fission scheme and its
parameters. The proofs are omitted, the methods being similar to the ones used
in [2].

For ease of discussion, we fix the following notations. Let $q$ be a prime
power and $X$ be the AST obtained from

$ASL(2,q)=\\{(x,y)^{T}\mapsto A(x,y)^{T}:A\in SL(2,q)\\},$

the group of affine transformations on the affine space $V=(GF(q))^{2}$ whose
linear parts have determinant 1. For $a\in GF(q)$, let $\vec{a}=(a,0)^{T}\in
V$. Additionally, for $(u,v,w)\in V\times V\times V$, let $[(u,v,w)]\in X$
denote the orbit of $(u,v,w)$ under $ASL(2,q)$. The size and third valencies
of $X$ are given in the following theorem.

######  Theorem 6.6.

Let $q$ be a prime power and $X$ be the AST obtained from the action of
$ASL(2,q)$. The two-point stabilizer $ASL(2,q)_{\vec{0},\vec{1}}$ has $2q-3$
orbits on $V\setminus\\{\vec{0},\vec{1}\\}$. There are $q-2$ orbits of the
form $\\{\vec{a}\\}$, where $a\neq 0,1$, and $q-1$ orbits of the form
$\\{(c,\mathfrak{a})^{T}:c\in GF(q)\\}$, where $\mathfrak{a}\neq 0$. Thus, $X$
has $2q-3$ nontrivial relations. There are $q-2$ nontrivial relations of the
form

$R^{a}=\\{[(\vec{0},\vec{1},\vec{a})]\\},\;a\neq 0,1,$

each with third valency 1. The remaining $q-1$ nontrivial relations of $X$ are
of the form

${}^{\mathfrak{a}}\\!R=\\{[(\vec{0},\vec{1},(0,\mathfrak{a})^{T})]\\},\;\mathfrak{a}\neq
0,$

each with third valency $q$.

For notational convenience, let $A^{a}$ denote the adjacency hypermatrix
corresponding to the relation $R^{a}$ whenever $a\neq 0,1$. Similarly, let
${}^{\mathfrak{a}}\\!A$ denote the adjacency hypermatrix corresponding to the
relation ${}^{\mathfrak{a}}\\!R$ whenever $\mathfrak{a}\neq 0$. The
intersection numbers of the subalgebra generated by the adjacency
hypermatrices of the nontrivial relations of $X$ are given implicitly in the
next theorem.

######  Theorem 6.7.

Let $q$ be a prime power and $X$ be the AST obtained from the action of
$ASL(2,q)$. The following equations hold for any $a,b,c\neq 0,1$ and
$\mathfrak{a},\mathfrak{b},\mathfrak{c}\neq 0$.

  1. (1)

$A^{a}A^{b}A^{c},=\begin{cases}A^{bc}&\text{if }bc=a(1-c)+c\neq 1,\\\
0,&\text{otherwise}.\end{cases}$

  2. (2)

$A^{a}A^{b}\,{}^{\mathfrak{c}}\\!A=A^{a}\,{}^{\mathfrak{c}}\\!A\,A^{b}={}^{\mathfrak{c}}\\!A\,A^{a}A^{b}=0$.

  3. (3)

${}^{\mathfrak{a}}\\!A\,{}^{\mathfrak{b}}\\!A\,A^{c}=\begin{cases}{}^{\frac{\mathfrak{b}}{c}}\\!A&\text{if
}\mathfrak{a}c+\mathfrak{b}c=\mathfrak{b},\\\ 0,&\text{otherwise}.\end{cases}$

  4. (4)

${}^{\mathfrak{a}}\\!A\,A^{c}\,{}^{\mathfrak{b}}\\!A=\begin{cases}{}^{{\mathfrak{b}}{c}}\\!A&\text{if
}\mathfrak{b}c=\mathfrak{a}+\mathfrak{b},\\\ 0,&\text{otherwise}.\end{cases}$

  5. (5)

$A^{c}\,{}^{\mathfrak{a}}\\!A\,{}^{\mathfrak{b}}\\!A=\begin{cases}{}^{{\mathfrak{b}}{(1-c)}}\\!A&\text{if
}\mathfrak{a}=-\mathfrak{b}c,\\\ 0,&\text{otherwise}.\end{cases}$

  6. (6)

${}^{\mathfrak{a}}\\!A\,{}^{\mathfrak{b}}\\!A\,{}^{\mathfrak{c}}\\!A=\begin{cases}qA^{-\frac{\mathfrak{b}}{\mathfrak{c}}}&\text{if
}\mathfrak{a}+\mathfrak{b}+\mathfrak{c}=0,\\\
{}^{\mathfrak{a}+\mathfrak{b}+\mathfrak{c}}\\!A&\text{if
}\mathfrak{a}+\mathfrak{b}+\mathfrak{c}\neq 0.\end{cases}$

The last theorem gives the intersection numbers $p_{ijk}^{l}$ of the ASTs
obtained from $ASL(2,q)$ whenever exactly one of $R_{i}$, $R_{j}$, and $R_{k}$
is trivial. Here $I_{1}$, $I_{2}$, and $I_{3}$ denote the respective adjacency
hypermatrices of the trivial relations $R_{1}$, $R_{2}$, and $R_{3}$ of $X$.

######  Theorem 6.8.

Let $q$ be a prime power and $X$ be the AST obtained from the action of
$ASL(2,q)$. The following equations hold for any $a,b\neq 0,1$ and
$\mathfrak{a},\mathfrak{b}\neq 0$.

  1. (1)

$I_{1}A^{a}A^{b}=\begin{cases}I_{1},&\text{if }ab=1,\\\
0,&\text{otherwise.}\end{cases}$

  2. (2)

$A^{a}I_{2}A^{b}=\begin{cases}I_{2},&\text{if }ab=a+b,\\\
0,&\text{otherwise.}\end{cases}$

  3. (3)

$A^{a}A^{b}I_{3}=\begin{cases}I_{3},&\text{if }a+b=1,\\\
0,&\text{otherwise.}\end{cases}$

  4. (4)

$I_{1}A^{a}\,{}^{\mathfrak{a}}\\!A=I_{1}\,{}^{\mathfrak{a}}\\!A\,A^{a}=A^{a}I_{2}\,{}^{\mathfrak{a}}\\!A=\,{}^{\mathfrak{a}}\\!A\,I_{2}A^{a}=A^{a}\,{}^{\mathfrak{a}}\\!A\,I_{3}=\,{}^{\mathfrak{a}}\\!A\,A^{a}I_{3}=0$.

  5. (5)

$I_{1}\,{}^{\mathfrak{a}}\\!A\,{}^{\mathfrak{b}}\\!A=\begin{cases}qI_{1}&\text{if
}\mathfrak{a}=\mathfrak{-b},\\\ {0},&\text{otherwise}.\end{cases}$

  6. (6)

${}^{\mathfrak{a}}\\!A\,I_{2}\,{}^{\mathfrak{b}}\\!A=\begin{cases}qI_{2}&\text{if
}\mathfrak{a}=\mathfrak{-b},\\\ {0},&\text{otherwise}.\end{cases}$

  7. (7)

${}^{\mathfrak{a}}\\!A\,{}^{\mathfrak{b}}\\!A\,I_{3}=\begin{cases}qI_{3}&\text{if
}\mathfrak{a}=\mathfrak{-b},\\\ {0},&\text{otherwise}.\end{cases}$

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