COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

224 Robert J. Plemmons
such class [3]. However, the computation time is then proportional to the
total number of such systems. The algorithm presented in this paper constructs
only those groupoids that are neither isomorphic nor anti-isomorphic;
and, after adding a routine to ensure associativity, it makes
feasible the construction of all semigroups of order N G 6, on most
modern digital computers.
The algorithm will be given for the most general binary system, the
groupoid, since routines to restrict the binary operations can readily be
added. Let N be a positive integer and let S be the set of all positive integers
less than or equal to N. We choose this as the set of elements for our
groupoids of orderN, since they are readily used as subscripts in FORTRAN.
Let RN denote the set of all NXN matrix arrays
A = (Uij)
where each aij E S. Then each array (ajj) represents a groupoid of order N
with binary operation o defined by
io j = OTij.
Conversely, each groupoid of order N has such a representation.
Let PM denote the permutation group on S and let AT denote the transpose
of A E RN. For each CL E PN and each A = (aii) E RN we define
Aa = B = (bij)
where
bij = a[a,-l(Ql-lcjj], i, j E S.
Then two groupoids of order N, represented by A and B in RN, are isomorphic
if and only if
Aa = B
for some a E PN, and are anti-isomorphic if and only if
AT/3 = B
for some B E PN.
Now the set RN is ordered by the relation G) defined by the rule that
ifA, BERNthen
AsB
if and only if aij = b, for each i, j E S, oL p _else
there is a pair tn, k E S such
that
‘&nk -== bmk
and aij = b, for all i, j E S where
j+(i-l)N -C k+(m-l)N.
In other words the ordering is row-wise.
Non-equivalent finite semigroups 225
Now for each A E RN we let IA denote the set of all Aa, as a ranges over
Phr. Then either IA = IA2. or else IA 17 1,, = 0, and moreover I, = ZAT if
and only if Aa = AT for some a E Pni.
In order to construct only those groupoids that are non-equivalent, we
construct only the minimal matrix (with respect to the ordering) in the
set I, (J IAT. This can be accomplished in the following way, using the familiar
backtrack method of exhaustive search. Starting with all ones (the null
groupoid), we initiate a process of backing up and going forward, row-wise,
in defining terms in the table, beginning with the last position, Now whenever
the process backs up in the table, the previous position is zeroed and
we consider the product there as undefined. In general, suppose the process
is at position (i, j). The term aij is replaced by a,+ 1. Then:
1. If Uij > N, we set aG = 0 and back up in the table. If i = j = 1,
the process is complete and we have all the desired systems; if not, the
process goes to position (i, j- 1) if j + 1 or (i- 1, N) if j = 1.
2. If aij