Properties of Semigroups Page: 43
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:
4$
y ° jPCy) for each, y in R(F). The above definitions of
5* and FF clearly imply F(x) = F(F(x)) since F « FF and
both F and FF are functions. (Thus y = F(y) for each y
in R(F). Therefore the theorem is proved.
Theorem 4.4. Let X be a set. Then F is a right zero
element of if, and only if, F is a constant transforma-
tion. Also, there are no left zero elements in 0X if X
is non-degenerate.
Proof. Let F be a constant transformation of X and
show F is a right zero element of Vx . Let GGtJx an<^- show
GF = F. Now F is a constant transformation of X implies
there exists an element a in S such that F = £(x,a) | x£ X^,
By the definition of the product of two transformations,
GF as £(x,F(G(x))) | x ^3 * Now G(x) £! X for every x in
X. Hence F(G(x)) = a and it is clear that GF = F. Thus
F is a right zero element of .
Let F be a right zero element of and show F is a
constant transformation of X. To show F is a constant
transformation of X it is necessary to show that the range
of F is a set consisting of a single element. Let b£ S
and consider g = £(x,b) | x £. X J . It is clear that
g£ *7x since g is a function whose domain is X and whose
range is fbl^X. Now GC^ and F is a right zero element
of imply GF = F. By definition,
GF = £(x,F(G(x))) | x £ x} and since G(x) ■ b for every
x£X, it follows that GF « £(x,F(b)) I x x} • Now
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/m1/46/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .