Math 20201: Algebraic Structures I (sections 1-5)

Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair of elements of elements of S a uniquely determined element of S. The element assigned to the ordered pair (a, b) with a, b ∈ S is denoted by a * b. Remark. In other words, a binary operarion of a set S is a function * : S×S → S from the Cartesian product S × S to the set S. The only difference is that the value of the function * at an ordered pair (a, b) is denoted by a * b rather than * ((a, b)). Definition. A binary operation * on a set S is commutative, if a * b = b * a ∀a, b, ∈ S. The binary operation ⋆ is commutative, but the binary operations ⋄ and are not commutative. Let * be a binary operation on a set S. and let a, b, c ∈ S. Consider the expression a * b * c. This expression doesn't have a meaning since * gives only a meaning to ordered pairs of elements. In fact, there are two ways of making a * b * c respectable, namely (a * b) * c and a * (b * c). For the operation ⋆ we have (3 ⋆ 2) ⋆ 4 = 3 ⋆ 4 = 4 3 ⋆ (2 ⋆ 4) = 3 ⋆ 4 = 4. In fact, for all a, b, c ∈ N we have (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c) = max(a, b, c).