Properties of Semigroups Page: 3
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Definition 1.8. Let each, of A and. B be a set. The
intersection of A and B is denoted by A A B and defined
by the following. If there exists at least one element
of A which is also an element of B, then A A B is defined
by: AHB = [x | x £ A and xC B] , If no such element
exists, then A and B are said to be mutually exclusive and
A AB has no meaning.
Definition 1.9. An ordered pair of elements, a and b,
is denoted by (a,b).
Definition 1.10. The statement that R is a.relation
means E is a set of ordered pairs.
Definition 1.11. The inverse of a relation R is de-
noted by R""^ and is defined by the following.
R""1 = {(x,y) [ (y,x) £. R } .
Definition 1.12. Let R be a relation. The domain
of R is denoted by D(R) and is defined by:
D(R) = ^x J (x,y) € R for some y} . The range of R is
denoted by R(R) and R(R) = DCR"1).
Definition 1.15. The statement that F is a function
means F is a relation in which no two ordered pairs have
the same first element. It will be convenient to use the
notation y = F(x) to mean that (x,y) £ F.
Definition 1.14. The statement that F is a reversible
function means both F and F~"^ are functions.
Definition 1.15. Let each of A and B be a set. Then
AXB ~ {(a,b) | a € A and b £ B} .
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Donnell, William Anthony. Properties of Semigroups, thesis, June 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130682/m1/6/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .