ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

4. 3: Tables with homomorphisms
In the following tables A, X, S, M, G, R, L, V are denoting an arbitrary structure
of the following types respectively general structure, set, semigroup, monoid,
group, ring, field.
If we have more structures of the same type we shall use indexed letters in the
usual way, so that M1, M2, . . . etc denote sets. We let M ′ denote a subset of
M and so on.
The set of homomorphisms of a structure A into a structure B is denoted by
Hom(A, B).
Homomorphisms wrt binary operations.
x ♢ y A x f(x) B u ♡ v symbo
1 x + y Rn x Ax Rm u + v
2 m + n Z n an R uv exp. m
3 A + B Mat(m, n, L) A L(A, V, W) Lin(V, W ) u + v ass. lin. m
4 L + M Lin(V, W ) L M(L, V, W) Mat(m, n, L) u + v ass. ma
5 LM Lin(V, W ) L M(L, V, W) Mat(m, n, L) u ◦ v ass. ma
6 P + Q L[X] P P (L) Lin(V, V ) u + v
7 x + y Rn x Ax Rm u + v mult. m.
8 xy N x σ(x) N0 N
u + v prime spec
9 xy OrdA x |x| N u + v word len
10 x × y Finite sets x |x| N0 uv ant. ele
11 x ◦ y Rot(3) x ˆx Mb u ◦ v ass. Möbius tran
12 x ◦ y Mb x ˆx Rot(3) u ◦ v ass. ro
13 xy SU(2) x ˆx SO(3) uv ass. ro
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5: Examples and exercises
Example 1: Different ways of notation. (see nr 5)
We have that +(a, ·(b, c)) = +a(·bc) = a + b · c = abc · +
Exercise 2: Backward polish. (see nr 5)
10 2 · 5 + √ 2 · 10 : = 1
Example 3: Linear mappings are homomorphisms (see nr 8)
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