Properties of Semigroups Page: 4
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Definition 1.16. Let S be a set. The statement that
0 is a "binary operation defined on S means 0 is a function
whose domain is SXS and whose range is a subset of S.
Definition 1.17. Let 0 be a binary operation defined
on a set S. The statement that 0 is associative means if
a € S, b<£ S, c C S, ((a,b) ,x) € 0, ((x,c),y) €. 0,
((b,c),z) £ 0, and ((a,z),w) €. 0, then y = w.
Definition 1.18. Let 0 be an associative binary
operation defined on a set S. The following notation will
be used. If ((a,b),x) €. 0, then x will be denoted by ab.
It follows that 0 is associative means if a£S, b €~ S,
and c £ S, then (ab)c = a(bc).
Definition 1.19. The statement that S is a semigroup
means S is a set on which there is defined an associative
binary operation.
Definition 1.20. The statement that a semigroup S is
of finite order means there exists a positive integer n
which corresponds to the number of elements in the set S.
Also, S is said to be of order n. If no such positive
integer exists, then S is said to be of infinite order.
Definition 1.21. The statement that a semigroup S
is degenerate means S is of order 1. The statement that
S is non-degenerate means there are at least two elements
in the set S.
Definition 1.22. Let each of S and S' be a semigroup.
The statement that S is isomorphic with S*, denoted by
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Donnell, William Anthony. Properties of Semigroups, thesis, June 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130682/m1/7/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .