Properties of Semigroups Page: 34
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34
regular implies there exists an element x in S such that
a = axa. Then a « axa implies xa = (xa)(xa). Thus xa is
idempotent. Now it remains to show L(a) = L(xa). To show
L(a)j=s L(xa), let y £ L(a) and show y£ L(xa). Now y<^ L(a)
implies y = a or y £ Sa. If y = a, then y = axa since
a = axa. Thus y = axa implies y £ Sxa since a(xa) £ Sxa.
If y £ Sa, then there exists an element y' in S such that
y « y'a. Then y = y'a and a = axa imply y = (y'a)xa which
implies y£ Sxa. Now y d Sxa implies y£"L(xa). Thus
L(a) L(xa). To show I(xa)£ 1(a), let z £ L(xa) and
show z £ L(a). Now z £ L(xa) implies z = xa or z £ Sxa.
If z = xa, then z £ Sa. If z £ Sxa, then there exists an
element z' in S such that z = z'za. This implies z £ Sa
since (z'z)a£ Sa. Then z £ Sa implies z£ L(a) which im-
plies . L(xa) L(a). Hence L(a) = L(xa) and xa is idempotent,
Suppose there exists an idempotent element e in S
such that L(a) = L(e) and show a is regular. The element
a is in L(a) and L(a) = L(e) imply a = e or a£. Se. If
a a e, then a = aaa since e is idempotent implies e = eee.
If a £ Se then there exists an element a' in S such that
a = a'e. Then a = a'e and e is idempotent imply a = ae.
Now e£ L(e) and L(e) = L(a) imply e = a or e £ Sa. If
e = a, then a aaa since e is idempotent. If e £ Sa, then
there exists an element e' in S such that e = e'a. Thus
a = ae and e = e'a imply a = ae'a. Therefore a is regular
and the theorem is proved.
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Donnell, William Anthony. Properties of Semigroups, thesis, June 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130682/m1/37/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .