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 : kiK). 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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