Modern Algebra Quiz 1

Operation

Let E be a non empty set. An operation on E is a rule # by which we associate to each ordered pair of elements of E exactly one element of E.

Composition

The element associated to the pair (x,y) is denoted by x # y and is called the composition of element x and y in that order

Grupoid

A nonempty set E with an operation is called a grupoid and is denoted by (E,#), or shortly by E

Addition

An operation denoted by +

sum of the addends

the composition x+y

Multiplication

An operation denoted by *

Product of the factors

the composition of x*y

Extension of the operation # to the power set of E

Let (E,#) be a grupoid. The operation on P(E), also denoted by #, defined in the following way: A#B = {a#b: a E A, b E B} for any A,B E P(E)

Stable under the operation # or Closed under the operation #

Let (E,#) be a grupoid. A nonempty subset of A of E is said to be stable under the operation # or closed under the operation #, if for any two elements a,b of A the composition a # b belongs to A.

The operation induced by the operation #

If a subset is stable under the operation # then we have the restriction of the operation # on A. We call that restricted operation the operation induced by the operation # on E

Subgrupoid

A subgrupoid of the grupoid (E,#) is a stable subset A of E with the operation induced by #

Stable subset of E generated by S

Let (E,#) be a grupoid and S a nonempty subset of E. The smallest stable subset <S> of E containing S is called the stable subset of E generated by S

Subgrupoid of E generated by S

When a stable subset of E generated by S is equipped with the induced operation, S is a generating set of <S>