《Discrete Mathematics II教学-华南理工》Lecture 3. Isomorphic Binary Structures.pdfVIP

I.3 Isomorphic Binary Structures 1 Section I.3. Isomorphic Binary Structures Note. In mathematics, an “iso-morphism” is, as the name suggests, a mapping which preserves structure. In a graph, the structure is connectedness. In a group (to be introduced in the next section), the structure is given by the binary operation. Note. Consider the tables: + 0 1 2 ∗ a b c ∗ x y z 0 0 1 2 a a b c x x y z 1 1 2 0 b b c a y z y x 2 2 0 1 c c a b z y x z Notice that the structure of operation + on {0, 1, 2} is the same as the structure of ∗ on {a, b, c}. This can be seen by replacing 0, 1, 2 with a, b, c (respectively) and + with ∗. Then any equation involving the first table yields an equation involving the second table (and vice-a-versa). So the only difference in binary operation + and binary operation ∗ is one of notation. So the first and second tables represent isomorphic binary operations (on the appropriate sets). However, the third table is fundamentally different—in it, the binary operation when applied to a pair of the same elements yields that element (it is an idempotent binary operation—see page 28, Exercise 2.37). This is not the case in the first two tables and so ∗ is not isomorphic to + nor ∗. Definition. A binary algebraic structure is an ordered pair S, ∗ where S is a set and ∗ is a binary operation on S . I.3 Isomorphic Binary Structures 2 Definition 3.7. Let S, ∗ and S , ∗ be binary algebraic structures. An isomor- phism of S with S is a one-to-one function φ mapping S onto S such that