## Chapter 5

## ALGEBRAIC STRUCTURES

##### 5.1 ALGEBRA

Algebra is tuple <*K, op*_{1} *op*_{1} *op*, …., *op _{n}*> where

*K*is a set called the “carrier” of algebra and

*op*

_{1}, ….,

*op*are operators. Each operator is a total function with domain

_{n}*k*for same

^{i}*i*and range

*k*(i.e.,

*op*:

_{j}*k*→

^{i}*K*). Thus,

*K*is closed under each of the operations

*op*.

_{j}**Remark** If *op _{j}*. has domain

*k*then

^{i}*op*is an operator of degree

_{j}*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.

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*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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