This chapter describes the function `SemiLatinSquareDuals' which can
classify semi-Latin squares with certain given properties, and return
a list of their duals as block designs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Semi-Latin squares and SOMAs}
Let $n$ and $k$ be positive integers. An $(n\times n)/k$ *semi-Latin
square* \index{semi-Latin square}
is an $n$ by $n$ array $A$, whose
entries are $k$-subsets of a $kn$-set $X$ (the *symbol-set*), such that
each element of $X$ occurs exactly once in each row and exactly once in
each column of $A$. (Thus an $(n\times n)/1$ semi-Latin square is the same
thing as a Latin square of order $n$.) For extensive useful information on
semi-Latin squares, see \URL{https://webspace.maths.qmul.ac.uk/r.a.bailey/sls.html}.
A SOMA$(k,n)$ \index{SOMA}
is an $(n\times n)/k$ semi-Latin square $A$,
with $n\ge2$, in which no 2-subset of the symbol-set is contained in
more than one entry of $A$. For extensive useful information on SOMAs,
see \URL{https://webspace.maths.qmul.ac.uk/l.h.soicher/soma/}.
Let $A$ and $B$ be $(n\times n)/k$ semi-Latin squares. We say that
$B$ is *(weakly) isomorphic* to $A$ if $B$ can be obtained from $A$
by applying one or more of: a row permutation; a column permutation;
transposing; renaming the symbols. If transposing is not allowed then we
get the concept of strong isomorphism. More formally, $B$ is *strongly
isomorphic* to $A$ if $B$ can be obtained from $A$ by applying one or
more of: a row permutation; a column permutation; renaming the symbols.
Let $A$ be an $(n\times n)/k$ semi-Latin square. Then the dual of $A$
can be represented as a binary block design as follows. The point-set of
$D$ is taken to be the Cartesian square of $\{1,\ldots,n\}$, with $[x,y]$
representing the $[x,y]$-entry of $A$. The blocks of $D$ are in one-to-one
correspondence with the symbols of $A$, with the $i$-th block of $D$
consisting of the ordered pairs $[x,y]$ such that the $i$-th symbol of
$A$ is contained in the $[x,y]$-entry of $A$. Given $D$, the semi-Latin
square $A$ can be recovered, up to the naming of its symbols.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{The function SemiLatinSquareDuals}
Let <n> and <k> be positive integers. Then this function (which makes
heavy use of the function `BlockDesigns') returns a list
of block
designs which are the duals of the $(<n>\times <n>)/<k>$ semi-Latin
squares whose properties are specified by the given parameters, described
below. In practice, depending on the specified properties, this function
can be useful for <n> up to about $6$ or $7$.
The parameter <maxmult>, if given, must be a positive integer or
the string `"default"'. If it is a positive integer, then
specifies an upper bound on the multiplicity of each block in each
semi-Latin square dual in <DL>. The default value for <maxmult> (if
omitted or if given as `"default"') is , which poses no constraint
on the block multiplicities.
The parameter <blockintsizes>, if given, must be a set of non-negative
integers or the string `"default"'. If it is given as a set, then
<blockintsizes> specifies, for each semi-Latin square dual in <DL>,
the set of possible sizes for the intersection of a block $B$ with a
different block (but possibly a repeat of $B$). The default value for
<blockintsizes> (if omitted or if given as `"default"') is `[0..]',
which poses no constraint on the block intersection sizes. Note that
block intersection sizes in the dual of a semi-Latin square correspond
to concurrencies of points in the semi-Latin square itself. Also note
that if $<n>\ge2$ and <blockintsizes> is specified to be `[0,1]' then
the $(n\times n)/k$ semi-Latin squares being considered are SOMA$(k,n)$s.
The parameter <isolevel>, if given, must be 0, 1, 2, 3, 4 or the string
`"default"' (the default value is 2). The value 0 specifies that
will contain at most one (semi-Latin square dual given as a) block design,
and will contain one such block design if and only if a semi-Latin square
with the required properties exists. The value~1 specifies that <DL>
will contain a list of duals representing all weak isomorphism classes
of semi-Latin squares with the required properties (possibly with some
classes represented more than once) and the value 2 specifies that <DL>
will contain precisely one dual semi-Latin square representative for
each weak isomorphism class of semi-Latin squares with the required
properties. The values 3 and 4 for <isolevel> play the roles of 1 and 2,
respectively, but with weak isomorphism replaced by strong isomorphism.
Thus, $<isolevel>=3$ specifies that <DL> will contain a list of duals
representing all strong isomorphism classes of semi-Latin squares with
the required properties (possibly with some classes represented more than
once) and $<isolevel>=4$ specifies that <DL> will contain precisely one
dual semi-Latin square representative for each strong isomorphism class
of semi-Latin squares with the required properties.
For example, we determine the numbers of weak and strong isomorphism
classes of $(4\times 4)/k$ semi-Latin squares for $k=1,\ldots,6$. (These
numbers disagree with P.~E.~Chigbu's classification for the cases $k=3,4$ \cite{BaCh}.)
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