Semigroup

An algebraic structure (S,*) is called a semigroup if it follows associative property (a*b)*c = a*(b*c) ∀a,b,c∈S

Example #

• (N,+): yes
  • It has the closure property, therefore it’s a algebraic structure.
  • (a+b)+c=a+(b+c) ∀a,b,c∈N: follows associative property
• (Z,*): yes
  • It has the closure property, therefore it’s a algebraic structure.
  • (a*b)*c=a*(b*c) ∀a,b,c∈Z: follows associative property
• (Z,-): no
  • It has the closure property, therefore it’s a algebraic structure.
  • (a-b)-c≠a-(b-c) ∀a,b,c∈Z: doesn’t follow associative property.
• (S={2ⁿ; n∈Z}, *): yes
  • 2ᵃ * 2ᵇ = 2ᵃ⁺ᵇ = 2ᶜ ∀2ᵃ,2ᵇ∈S and also 2ᶜ∈S: follows closure property.
  • (2ᵃ*2ᵇ)*2ᶜ=2ᵃ*(2ᵇ*2ᶜ) ∀a,b,c∈S: follows associative property.