A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids. A semigroup may have at most one two-sided identity.

What is semigroup example?

A motivating example of a semigroup is the set of positive integers with multiplication as the operation. for all x and y in S. Commutative semigroups are often written additively. A subsemigroup of S is a subset T of S which is closed under the binary operation and hence is again a semigroup.

What is monoid example?

If a semigroup {M, * } has an identity element with respect to the operation * , then {M, * } is called a monoid. For example, if N is the set of natural numbers, then {N,+} and {N,X} are monoids with the identity elements 0 and 1 respectively. ... The semigroups {E,+} and {E,X} are not monoids.