Properties of Semigroups Page: 6
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element of S. The statement that S is commutative means
if each of a and b is in S, then ab = ba.
Theorem 1.1. If S is a semigroup of finite order,
then S contains an idempotent element.
Proof. The semigroup S is of finite order implies
there exists a positive integer j which corresponds to
the number of elements in S. ( Now let a C S and let
A = {a, a2, a^,..., a0', a^+1] . It is clear aQs since
a S and S is a semigroup. Now there are o+l representa-
tions for the elements in A and since ASs and S contains
only j elements, it follows that there exist positive
jd. 1c
integers n and k where n<k<£j+l and such that a = a .
Now n<k implies there exists a positive integer p such
that k = n+p. Thus a11 = ak and k = n+p imply a11 = an+I>.
The statement that a11 = an+m]? where m is a positive integer
will be proved by mathematical induction. The statement
is true for m = 1 since a11 = arL+^ implies a11 = aI1+^.
Now assume a11 an+tp is true for the positive integer
t and show a11 = an+(t+l)p ^s true. Clearly a11 = &n+^
implies a11 = (an)(a^). Now a11 = (an)(a^) and a11 = an+^
imply a11 = (an+-^)(a^) which implies a11 = an+-^+^ which
implies a11 = an+(t+l)p^ Tiru.s &n = an+mp ig ^rue for eacll
positive integer m and, in particular, a11 « an+I1-^. If
p = 1, then a11 = an+nP arL£ p = 1 imply a11 = an+n which
implies a11 = a211. If p>-l, then a11 = an+11P and.
anp-n = &np-n an(anP~n) = an+nP(anP~n) which implies
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Donnell, William Anthony. Properties of Semigroups, thesis, June 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130682/m1/9/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .