Monoid

A semigroup (S,*) is called a monoid if there exists an element e∈S such that (a*e)=(e*a)=a ∀a∈S Element ’e’ is called identity element of S wrt ‘*’

Example #

• (Z,+): yes
  • It is a semigroup because it has the properties of closure and associativity.
  • a+e=a ∀a∈Z, then we know that e=0. There exists an identity element of Z wrt ‘+’
• (N,*): yes
  • It is a semigroup because it has the properties of closure and associativity.
  • a*e=a ∀a∈N, e=1. There exists an identity element of N wrt ‘*’
• (R,÷): no
  • It is not an algebraic structure because real numbers are not closer under division (cuz of 0)
• (R*,÷): no
  • It is an algebraic structure but not a semigroup because it doesn’t follow associative property
• (S={2ⁿ; n∈Z}, *): yes
  • We know it’s a semigroup because it has the properties of closure and associativity.
  • 2ⁿ*2ˣ=2ⁿ ∀n∈Z. 2ˣ=1 => x=0∈Z. There exists an identity element of S wrt ‘*’