We remark that the following definitions given here, are either
identical, or basically the same as the corresponding definitions used
by Soicher in SOMA
update
n × n
array A, whose
entries are k-subsets of a kn-set Ω (the symbol-set),
such that
every symbol of Ω occurs exactly once in each row and exactly
once in
each column of A, and every 2-subset of Ω is contained
in most one entry of A.A SOMA Let - for any
*i,j∈{1,2,...,n}*, there exists exactly*k*elements of*B*that map*i*to*j*, and - for any distinct
*a, b ∈ B*, there exists at most one*i∈{1,2,...,n}*such that*ia=ib.*
kn-set of permutations.
## The structure of SOMAsLet B
if, and only if, C
itself is a SOMA(k',n) for some integer k'. Defintion: A decomposition of
B is a partition {B
of _{1},...,B_{m}}B,
such that each
B is a subSOMA_{i}(k of _{i},n)B,
for some integer k . We
then call the sequence _{i}(k a _{1},..,k_{m})type
of B. It
is clear that
B, such
that each B is an indecomposable subSOMA_{i}(k
of _{i}
,n)B, for some integer k .
We then call the sequence _{i}(k
a _{1},..,k_{m})unrefinable
decompostion type (ud-type)
of B.## Graphs corresponding to SOMAsGiven a SOMA We remark that two SOMA |