COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

252 Takayuki Tamura
Let U be a subset of SFT, and define
Prj,(U) = {a E T; (~3 a> E U}.
Dejinition. If U is a subsemigroup of Sy@T and if p*j,( U) = T, then U is
called a sub-general product of SXoT.
In the following theorem, the latter statement makes the theorem have
sense.
THEOREM 2.6. If a semigroup D is homomorphic onto a semigroup T under
a mapping g, then D is restrictedly isomorphic into SY@T with respect to g
and the projection of SEoT to Tfor some S. Furthermore there exists an SO
among the above S such that 1 S o j is either the minimum of 1 S/ or possibly the
minimum plus one.
Proof. Let D = U D,, D,g = a, Clearly 1 D,] e 1 D j for all a E T. The set
=ET
{lDol[; a E T} has a least upper bound. (For this the well-ordered principle is
used.) Let
m = 1 f1.u.b. {/DE 1; a E T}
and take a system of sets S, of symbols such that
1 S, 1 = m for all a E T
and a set So with I So I = m. Further we assume that D, & S, and S, contains
a special symbol 0,,
0, t Dw
and So contains a special symbol 0. Now let fb be a bijection of S to S, such
that
Of, = 0,.
We define a binary operation on G = SX T as follows:
(x, a)(y, j) = ~~~$$xG1~ aI@ ~~f2ra.s-$EDp
i
Then we can prove that G = SFoT where
K~f,)(Yfs)>f$
XQ9Y =
xf=EDg, YfpED,
{ 0 otherwise.
Let D’ = {(x, a); xf, E D,, a E T). Then Prj, (D’) = T and D’ 2 D under
b, a> -, xf,, a E T.
2.6. Construction of some general products. As a simplest interesting
example of general product, we will construct all general left products of
a set S by a right zero semigroup T.
ra B
Let T = {a, B), a a j
B a B
The equations (2.9) are
Semigroups and groupoids
e cc. cl* es. = 8,. %a 6,.
6. a% 8,. = e,. +kc, 0,.
8 L1. 0 j+ 8,. = 8,. +ka 0,.
%I+ (I++ e,. = eB. +kc, 0,.
253
(2.11)
8,. and %,. are semigroup operations. In order to construct all left general
products G = S??,T we may find all ordered pairs of semigroup operations
on S:
(%a., RJ.>
which corresponds to
G = G&L) U GK’$.>, ( G, 1 = 1 GB /.
For fixed T and S, G is denoted by G(%,., %,.). Clearly
G(%,., %,.) z G(%,., %a.).
Instead of ordered pairs it is sufficient to find pairs (%,., %@,) regardless of
order.
Let G,s denote the set of all semigroup operations defined on S. (Gs contains
isomorphic ones.) We define a relation w on Gs as follows:
%-~ifandonlyif%a~~=%~Aca~and~.~%=~~cn%forallaES.
The relation N is reflexive symmetric.
Let YE, y: be transformations of S defined by
zcp: = zex, zv: = X~Z
respectively. Then
8 =+++~cy = %+#, 7 for all a E S, if and only if
y&p; = @y$ for all x, y E S.
As special cases we will determine the relation N on Gs in the case
/SI=Z3.
I. Left generalproduct of S, ISI = 2, by right zero semigroup T.
a b
Let S = {a, b}. m , x, y, z, u = a or b, is the table a x y
b,z ru
Explanation of the notations which will be used later: For example
4 denotes the semigroup abl
b a j , i.e. 40 = 4.
4i denotes lbai i
abi
which is the isomorphic image of 4 under 1