Free Groups
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چکیده
In this note we give an alternative construction of free groups by working with semigroups. We also give some basic properties of free groups. A semigroup is a nonempty set with an associative binary operation. If S is a semigroup, we denote the operation by (a, b) 7→ ab. If S has an identity, then we call S a semigroup with 1. Such semigroups are often called monoids. We will restrict our attention to semigroups with 1, a minor assumption. A semigroup homomorphism is a function φ : S → T between semigroups such that φ(ab) = φ(a)φ(b) for all a, b ∈ S. We construct the free semigroup on a set. Let X be a set, and let S(X) be the (disjoint) union
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