Chapter 5
ALGEBRAIC STRUCTURES
5.1 ALGEBRA
Algebra is tuple <K, op1 op1 op, …., opn> where K is a set called the “carrier” of algebra and op1, …., opn are operators. Each operator is a total function with domain ki for same i and range k (i.e., opj : ki → K). Thus, K is closed under each of the operations opj.
Remark If opj. has domain ki then opj is an operator of degree i.
If i = 1 then it is an anon operator.
If i = 2 then it is a binary.
If i = 3 then it is a ternary.
- | K | is the order of algebra.
- K is a finite set then algebra is a finite algebra.
Example 1 Binary algebra <K, *> where * is a binary operator.
Semigroup <k, *> where * is associative.
Monoid <k, *> iff * is a binary associative operator with identity.
Definition Ring is ...
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