COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

226 Robert J. PIemmons
of A. Now suppose that
is the set of all distinct diagonals with the property that Di < Dia for
each u E Phi. Let
denote the set of all non-equivalent groupoids with diagonal Di. Then
LDi
is the set of all non-equivalent groupoids of order N. Thus to construct
these groupoids we need only construct each diagonal Di, lowest in the
ordering, determine the subgroup of PN leaving Di fixed, and then construct
all non-equivalent groupoids with diagonal Di by using this subgroup
in the equivalence checks. This procedure reduces the work factor
considerably. In addition to this, terms can often be defined ahead in
the table to ensure that the binary operation in the partial table has the
desired properties.
This algorithm has been coded into FORTRAN and the program has
been run, in one form or another, on several computers, including those at
Auburn University, the University of Tennessee and the National Security
Agency. Programs resulting from the algorithm have been applied to the
construction of various types of systems, such as groupoids, semigroups
and loops; with the construction of the Cayley tables for all semigroups
of order N 4 6 being one of the more noteworthy results. A monograph
listing these tables, along with other information, can be obtained from the
Department of Mathematics at Auburn University, Auburn, Alabama [8].
3. Some analysis results. These finite semigroups, constructed by use
of the algorithm described in Q 2, have been classified according to several
properties, such as being regular, inverse or subdirectly irreducible. Such
classification is accomplished by adding appropriate subroutines to the
construction program. A table giving the number of (regular, inverse)
semigroups of order N < 6 is given at the end of the paper (p. 228). Also
included is the number of semigroups containing k idempotents for
k = 1, . . ., N.
Perhaps the most interesting use of the algorithm has been in the construction
of specific finite systems of order N that satisfy certain identities
or have certain properties, or else proving that no such systems exist for
that order. For example, one such application has solved the problem of
finding the smallest order semigroup whose system of identities has no
finite basis [5], [6], [7].
THEOREM. There is a semigroup of order 5 whose system of identities has
no finite basis, and, moreover, the system of identities for each semigroup
of order N < 5 has a finite basis.
Non-equivalent finite semigroups 227
Other analysis results were obtained by constructing all the congruence
relations on each semigroup of order N G 5 and on selected semigroups
of order 6. The algorithm to construct these relations first determines the
equivalences on the set and then tests for compatibility with the binary
operations of the non-equivalent semigroups of that order. The resulting
examples are useful in the study of semigroup decompositions and in the
study of homomorphisms, since the consideration of homomorphisms can
be limited to the consideration of congruences. These computations have
suggested the next result.
THEOREM. The following four conditions concerning a semigroup S of
order N w 2 are equivalent.
(A) Each reflexive relation on S is left compatible.
(B) Each equivalence relation on S is a left congruence.
(C) For each x, y and z in S either xy = xz or xy = y and xz = z.
(D) S = A ~.j B, where An B = 0 and where, for some idempotent
function ffrom A to A, the binary operation for S is given by
If(x) if xEA.
xy =
I y if xc B.
This theorem, together with its dual, shows that each equivalence relation
on a semigroup S is also a congruence relation if and only if S is a [left,
right] zero semigroup.
The examination of these examples has also led to the following conjecture
:
If a finite semigroup of order N > 3 has exactly one proper congruence
relation, then it is a group or a simple group with zero.
In conclusion we mention that the construction of all non-equivalent
semigroups of order 7 would be rather difficult, both from the standpoint
of running time on any particular computer and from the standpoint of
output volume. Although there is no known rule giving the number of
non-equivalent finite semigroups as a function of the order, a good estimate
for the number of order 7 is around 200,000. However, the construction
and analysis of special types for N Z= 7 is sometimes feasible and
could be useful in the formulation and testing of conjectures.
REFERENCES
1. A. H. CLIFFORD and G. P. PRESTON: The Algebraic Theory of Semigroups, Amer.
Math. SOL, Math. Surveys, Vol. I (Providence, 1961).
2. A. H. CLIFFORD and G. P. PRESTON: The Algebraic Theory of Semigroups, Amer.
Math. Sot., Math. Surveys, Vol. II (Providence, 1967).
3. G. E. FORSYTHE: SWAC computes 126 distinct semigroups of order 4. Proc. Amer.
Math. Sot. 6 (1955), 443-445.