COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

248 Takayuki Tamura
Immediately we have :
PROPOSITION 2.1. Bn is a semigroup with respect to a +# and G++ (I for all a E E.
The semigroups ‘BE with n % and ++ Dare denoted by BE (,+#) and BE (s .)
respectively.
E(0) is associative if and only if 8 c1 + 8 = 8 + II 0 for all a E E.
Let v be a permutation of E. For 0 E BE, @ is defined by
ewy = ~b+xxY~-31~. (2.3)
Thus 9 induces a permutation of BE. For 8 E 5& another operation 0’ is
defined by
xefy = yex.
LEMMA. (0 (I +i+ 59~ = ed ag, * (w,
(0 gc, T)V = w M,, hd,
(e D ++ d’ = 7’ +tb et,
(e +kc, 7)’ = 7jl.g 0’.
PROPOSITION 2.2. Bnd SC) is isomorphic with BE6+c) and is anti-isomorphic
with ‘&( %t,) for all a, b E E.
2.3. General product. Let S be a set and T be a semigroup. Suppose
that a mapping @ of TX T into !-&, (a, p)O = 0,. B, satisfies
8z,p .g eaS,r = e,,, sc, eaBr for all a, p, y E T and all a E S. (2.4)
Consider the product set
SXT= {(x,a);xES,aET}
in which (x, a) = (y, ,8) if and only if x = y, a = /?.
Given S, T, 0, a binary operation is defined on SX T as follows
(x3 4 (Y, P> = was BY, aP>. (2.5)
PROPOSITION 2.3. SXTis a semigroup with respect to the operation (2.5),
and it is homomorphic onto T under the projection (x, a) +-a.
Definition. The semigroup, SXT with (2.5), is called a generalproduct of
a set S by a semigroup T with respect to 0, and is denoted by
SCOT.
If it is not necessary to specify 0 it is denoted by
S-R-T.
PROPOSITION 2.4. Suppose that T is isomorphic with T’ under a mapping
p and ISI = ISI; let v be a bijection S-S. Then
=oT g S’XolT’
where @= {e,,,da,B>ETXT}, 0’={e~~,,,;(ay,By)ET’xT’}
and xe:v,,,Y = [(~9-9~,,so~-3~91, X, Y E s’.
Semigroups and groupoids 249
In this case we say that 0 in S is equivalent to 0’ in s’.
We understand that SX oT is determined by T, ISI and the equivalence
of 0 in the above sense.
Dejinition. If a semigroup D is isomorphic onto some SXeT, then D is
called general product decomposable (gp-decomposable). If ISI w 1 and
ITI =- 1, then D is called properly gp-decomposable.
DeJinition. Let g be a homomorphism of a semigroup D onto a semigroup
T: D = UDW, D,g = a. If ID,1 = \DBl for all a, /? E T, then g is called
C&T
a homogeneous homomorphism (h-homomorphism) of D, or D is said to be
h-homomorphic onto T. If ID,1 =- 1 and IT/=- 1, then g is called a proper
h-homomorphism.
THEOREM 2.1. A semigroup D is gp-decomposable if and only if D has an
h-homomorphism.
In other words, D z SxoT, ISI w 1, ITI =- 1, for some 0 if and only if D is
properly h-homomorphic onto T.
Proof. Suppose that D is h-homomorphic onto T under g.
D = U D,, D,g = a.
=CT
Let S be a set with /SI = ID,1 for all a E T, and let fz be a bijection of D, to
S. Fixing { fE ; a E T}, for each (a, /?) E TX T we define a binary operation 8, B
on S as follows. Let x, y E S:
xk, p~ = [(xf3(xfi9if,p
Let a be any element of D, hence a E D, for some a E T. We define a mapping
y of D onto SX T as follows :
a 2 (af=, 4.
Then v is an isomorphism of D onto SxaT. The proof of the converse is
easy.
Even if D, S, Tare given, 0 depends on the choice of {f,; a E T}. However,
0 is unique in some sense. To explain this situation we shall define a
terminology.
Definition’Let gand g’ be homomorphisms of semigroups A and & onto a
semigroup C respectively. An isomorphism h of A into (onto) B is called a
restricted isomorphism of A into (onto) B with respect to g and g’ or we say
A is restrictedly isomorphic into (onto) B with respect to g and g’ if there is
an automorphism k of C such that h *g’ = g* k: