Properties of Semigroups Page: 38
iii, 48 leavesView a full description of this thesis.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
38
"be idempotent elements of S and show gh = hg. The element
gh is in S and S is an inverse semigroup imply there exists
a unique element x in S such that gh = ghxgh and xghx = x.
Clearly gh(hx)gh = gh and hx(gh)hx hxghx = hx. Thus
x = hx since x and hx are both inverses of gh. Similarly
it can be shown that x = xg. Now xx = xghx = x(gh)x = x
which implies x is idempotent. The element x is idempotent
implies x is its own inverse. Thus it follows that x = gh
since both x and gh are inverses of x. Thus gh is idempotent.
Similarly it can be shown that hg is idempotent., Now since
g, h, gh, and hg are idempotent elements, then, by Theorem
3.12, gh and hg are inverses of each other. Thus gh is its
own inverse since gh is idempotent. Thus gh = hg and (1)
is true. Therefore the theorem is true.
Theorem 3.15. If e and f are idempotent elements of
an inverse semigroup, then SeASf = Sef and Sef = Sfe.
Proof. Let &.Q Sef and show a € SeOSf. The element
a is in Sef implies there exists an element b in S such that
a = bef. It follows from Theorem 3*14- that ef = fe since
e and f are idempotent elements of an inverse semigroup.
Thus a = bfe. Now it is clear that a C Se and a £ Sf since
a = bfe and a = bef. Thus aC SeHsf. Hence Sef QSe/^Sf.
Prom the preceding part of the proof it is clear that
SeASf is meaningful. Thus let a €1 SeASf and show a Sef.
Then a G Se and a €1 Sf. The element a is in Se implies
there exists an element x in S such that a = xe. Clearly
Upcoming Pages
Here’s what’s next.
Search Inside
This thesis can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Thesis.
Donnell, William Anthony. Properties of Semigroups, thesis, June 1966; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc130682/m1/41/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .