COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

254 Takayuki Tamura
TABLE 5. Semigroups of Order 2
11 is exactly the same as 1, i.e. lo = 11.
1’ denotes
%Z ’
n
omitted from Table 5.
Table 6 shows all r] such that 8,, - 7. We may pick 13~ from all non-isomorphic
semigroups, but must select 7 from all semigroups. Generally the
following holds :
8 - 7 implies 8g, r~ 7~ for all permutations v of S (see 9 2.2), (2.12)
e - 7 implies +j’ - 8’. (2.13)
TABLE 6
e.
1
1’
2
3
4
rl
1
1’
2, 3
2, 3
4, 4,
From the table we also have
e. = 21, 7 = 21, 31,
e. = 31, r = 21, 31,
e. = 41, ?j = 4, 41.
Semigroups and groupoids 255
Table 7 shows all non-isomorphic left general products D of S, 1 S 1 = 2,
by a right zero semigroup of order 2.
TABLE 7
6 cz. %
1 1
1’ 1’
2 2
2 3
3 3
4 j 4
As an application of the above results, we have
THEOREM~.~. Let Sbeaset, IS/ = 2, andTbearight zerosemigroup of order
n. A left general product D of S by T is isomorphic onto either the direct
product of a semigroup S of order 2 and a right zero semigroup T of order n
DrSXT, lSl=2, jTl=n
or the union of the two direct products
D = (Sl X TI) U (Sz X T2),
where TI and T2 are right zero semigroups, / TII + I T2/ = n and SI is a null
semigroup of order 2 and S2 is a semilattice of order 2.
II. Left generalproduct of S, ISI = 3, by right zero semigroup.
a B
Let T = {a, ,4}, cc cc /? .
B.u B
The method is the same as in case I, and we use the same notation. Let
Gs denote the set of all semigroups defined on S, [ S/ = 3. Table 8 shows
Ga except the dual forms. Those were copied from [8], [lo]. Table 9 shows
all 17 for given B. such that B. IV 7.
This table shows, for example, that 2 = 20 = 21, 22 = 23, 24 = 25.
In the following family X of ten subsets of Gs, each set satisfies the
property: Any two elements of each set are --equivalent, and each set is
a maximal set with this property.
z
{I}, (2, 3, 15>, (4, 52, lb}, 6, 62, 64}, (7, 72, 1 l},
1 (7, 122}, (2, 8, 14’, 14;}, (2, 9, 18}, (2, 10, 101}, (2, 13, 131 17).
Let S’ denote the family obtained from S by replacing (6, 62, 64,) by (6)
and (2, 10, 101) by (2, lo} and leaving the remaining sets unchanged.