From sherman@cs.umbc.eduFri Nov 6 11:23:10 1998
Date: Fri, 6 Nov 1998 08:44:01 -0500 (EST)
From: "Dr. Alan Sherman"
To: "CMSC-641 Students
Subject: morphisms
A homomorphism (of an algebraic structure--such as a group) is
a function h : A -> B that "preserves" the algebraic structure
of A in the sense that
for all x,y in A, h( x * y ) = h(x) @ h(y),
where * is the (group) operation in A and @ is the (group) operation in B.
The noun "homomorphism" comes from the prefix "homo" (same) and the
suffix "morphism" (structure). Thus, a homomorphism preserves structure.
There are many special types of homomorphisms. For example, some
common homorphisms are:
monomorphism: injection (one-to-one)
epimorphism: surjection (onto)
isomorphism: bijection (injection + surjection)
endomorphism: A = B
automorphism: isomorphic endomorphism.
Sometimes, mathematicians use these terms (as I did in my abstract)
without necessarily implying any structure-preserving properties.
Since there is no more general term to express A=B (as surjection
generalizes epimorphism), I often use the noun endomorphism
in the sense of "Source = Target."
In addition, there are many other types of structure-preserving
functions (diffeomorphism, K-homomorphism, anti-isomorphism,
homeomorphism, inner/outer automorphism, ... )
Alan T. Sherman
sherman@umbc.edu
P.S. A note on the nouns "Source" and "Target." Most people
and books use the terms "function," "domain," "range," and "image,"
rather loosely. I dislike this practice. This practice results
from (a) sloppy thinking, and (b) awkwardness of being precise.
To me, a function is a triple F = (A, B, f), where
A is the source, B is the target, and f is the rule of assignment.
The expresson "Let F be the function f : A -> B" means
"Let F = (A, B, f) be a function." For convenience, authors
often (typically) sloppily write "f" in the sense of "F," in much
the same way that authors often write the set "G" in the sense of
the group "{\cal G}" = (G, +, 0).
A "rule of assignment" is any relation f \subseteq AxB that
satisfies the following "functional property:"
for all x,y1,y2, ( (x,y1) \in f ) and ( (x,y2) \in f ) implies y1 = y2.
[Compare with "injective":
for all x1,x2,y, ( (x1,y) \in f ) and ( (x2,y) \in f ) implies x1 = x2.
]
There are many equivalent notations for describing functions and their
values. For example, the most common notations are
y = f(x) (most common)
(x, y) \in f (I like when being very explicit and precise because
the full expressive power of set notation is available.)
y = fx (I dislike)
y = xf (I dislike, except when performing calulations on a
calculator with "Reverse Polish" notation.)
Sometimes the function is implicit and not mentioned by name.
for example, if g is a binary function g: AxB -> C, then
one might write a = bc to mean ( (a,b), c ) \in g.
Now,
Domain(f) = { x : for some y, (x,y) \in f }.
Image(f) = { y : for some x, (x,y) \in f}.
The following standard terms can now be easily and precisely
defined:
total: Domain = Source
partial: Domain \subseteq Source
strictly partial: Domain \subsetneq Source
surjective: Image = Target
I do not use the ambiguous noun "range" because some people define
range to be image, while others define range to be target. I recommend
that you avoid the ambiguous term range.
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