ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

holds for all a, b, c ∈ A
29. Theorem: Parentheses are redundant for associative operations
Easy to understand what it means but clumsy to formulate precisely.
30. Definition: Commutative operation
A binary operation ♢ on A is said to be commutative if
a ♢ b = b ♢ a
holds generally, that is for all a, b ∈ A
31. Definition: Distributive operation
En binary operation ♢ is said to be distributive wrt a binary operation ♡ if
a ♢ (b1 ♡ b2) = (a ♢ b1) ♡ (a ♢ b2)
holds generally, that is for all a, b ∈ A
32. Theorem: Associativity, commutativity and distributivity is hereditary
If an operation ♢ on A is associative, the same holds for the operations induced
on subsets, function spaces, product spaces and quotient spaces. The same holds
commutativity and distributivity.
Proof : An easy, may be tedious exercise.
33. Theorem: Associativity, commutativity and distributivity is preserved
by homomorphisms
Let ♢ be an operation in A and ♡ an operation in B and let f be a homomorphism
of A into B. If f is surjective and ♢ is associative then ♡
is associative. If f is injective and ♡ is associative then ♢ is associative.
Obvious analogues hold for commutativity and distributivity.
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