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. ---------------------------------------------------------------------------