COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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Takayuki Tamura Semigroups and groupoids 247
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PART II. SYSTEM OF OPERATIONS AND
EXTENSION THEORY
2.1. Introduction. Let T be a right zero semigroup, i.e. CC/? = ,!? for all
a, /? E T, and {D,; a E T} be a system of semigroups with same cardinality
ID,] = m. The p ro blem at the present time is to construct a semigroup D
such that D is a set union of D,, a C T, and
Dc,Dp C D,g for all a, /I E T.
D does not necessarily exist for an arbitrary system of semigroups. For
example, let
a b
D,: right zero semigroup of order 2. a a b
b ab
c d
D,: a group of order 2. c c d
d dc
Here D, 0 DZ = 0. Then there is no semigroup D satisfying
D = &UPz, &Dz S D2, D2Dl G DI.
So our question is this:
Under what condition on {D,; cc E T} does there exist such a semigroup
D?
How can we determine all D for given T and {D,; a E T}?
The problem in some special cases was studied by R. Yoshida [Ml, [19]
in which he did not assume the same cardinality of D,. In this paper we look
at the problem from the more general point of view; we will introduce the
concept of a general product of a set by a semigroup using the system of
groupoids. Finally we will show the computing results on a certain special
case. The detailed proof will be published elsewhere.
2.2. The system of operations.+ Let E be a set and BE be the set of all
binary operations (not necessarily associative) defined on E. Let x, y E E,
f3 E BE and let xOy denote the product of x and y by 8. A groupoid with 0
defined on E is denoted by E(0). The equality of elements of BE is defined
in the natural sense:
0 = 7 if and only if xey = xqy for all x, ycE.
Let a c E be fixed. For a we define two binary operations (I G# and +C a as follows
:
40 a++ 7)~ = WahY, (2.1)
CPA 17
xv sAc, 7)~ = x@yY). (2.2)
7 The system of semigroup operation was studied in [7].