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 ...
Get Discrete Mathematics by Pearson now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
