ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Proof : Associativity is inherited by the induced structures, and no more is
needed.
7. 3: Powers
The notion of power with positive integer exponent can be defined and the
power rules proved at this very primitive level. And then we have dealt with
this aspect once and for always. Notice that the definitions and the proofs are
the ones you learned very early in your life.
65. Definition: Powers with positive exponent.
In a semigroup parentheses are superfluous and so powers with positive integer
exponent of an element a may be immediately defined by the following recursive
procedure: a 1 = a, a n+1 = aa n
66. Theorem: Rules for calculating with Powers
a n a m = a n+m , (a n ) m = a mn for all a and all n, m > 0.
If ab = ba then (ab) n = a n b n .
Proof : Induction. Example. Let m be fixed. We show the formula for this m
and all n by induction after n.
For n = 1 we have u n u m = uu m = u m+1 per definition of power.
Suppose then that you have proved the formula for n = k. Then we have that
u k+1 u m = uu k u m = uu m+k = u m+k+1 , and so the formula is also proved for
n = k + 1.
8: Monoids
8. 1: Definitions
A very large part of the binary operations, used in algebraic structures, have a
neutral element, an element without action when used by the operation.
To study the pure effect of introducing a neutral element we introduce the
notion of a monoid, a structure which differs from a semigroup only by admitting
a neutral element. Actually there are a lot of interesting monoids without
further structure. For the sake of completeness we take the following
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