Given a set S and an operation +, we say that (S, +) is a semigroup if it satisfies the following properties for any x, y, z ∈ S:

  • Closure: x + y ∈ S
  • Associativity: (x + y) + z = x + (y + z)

We also say that S forms a semigroup under +.

Examples of Semigroups

  • Strings form a semigroup under concatenation.

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