In this chapter we shall study different algebraic structures with one binary operation and the relationship amongst them. The simplest algebraic structure is a groupoid. We begin with a few definitions.

**Definition 5.1**. *A non-empty set G equipped with a binary operation * is called a groupoid, that is, a * b ∈ G ∀ a, b ∈ G. This is also referred to as: G is closed with respect to **.

**Definition 5.2**. *A non-e
mpty set G equipped with a binary operation * is called a semigroup if * is associative, i.e.*

a * (b * c)=(a * b) * c ∀ a, b, ∈ G. (5.1)

**Definition 5.3**. *A non-empty set G equipped with a binary operation * is called a monoid if*

- * is associative, i.e.
a * (b ...

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