Algebra 2 | Semigroups

Semigroups are built from the simplest ingredients: a set S together with a binary operation ◦ that combines any two elements of S into another element of S. The key requirement is closure—whenever a and b lie in S, the result a ◦ b must also lie in S. From there, the central upgrade over a generic binary operation is associativity, which guarantees that parentheses can be dropped when combining three (or more) elements.

A binary operation is any function F: S × S → S, but it’s usually written in operation form like a ◦ b (or sometimes with symbols such as ⋆, multiplication, or +, depending on the context). For finite sets, an operation can be fully specified by an operation table listing the output for every ordered pair (a, b). Importantly, order can matter: in general, a ◦ b need not equal b ◦ a, and different parenthesizations of a ◦ b ◦ c can produce different results.

Associativity is the property that prevents that parenthesis problem. For all a, b, c in S, associativity requires (a ◦ b) ◦ c = a ◦ (b ◦ c). When this holds, the operation becomes “parenthesis-free” in practice: the overall order of the elements stays the same, but the grouping no longer changes the outcome. That’s the defining feature of a semigroup.

Formally, a semigroup is the pair (S, ◦) where S is a set and ◦ is an associative binary operation on S. The video emphasizes that this structure isn’t complicated—there are only two components, and the whole point is enforcing associativity on top of closure.

A concrete example comes from functions. Take the set S to be all functions from R to R. Define the binary operation ◦ as function composition: for functions f and g, the composition f ◦ g is the function that applies g first and then f. The operation is closed because composing two functions R → R always produces another function R → R.

To check associativity, pick three functions F1, F2, F3. Consider two ways to compose them: (F1 ◦ F2) ◦ F3 and F1 ◦ (F2 ◦ F3). Evaluating either composite at an arbitrary real number x shows both constructions yield the same output: the nested application of F3 to x, then F2, then F1, occurs in the same order regardless of how parentheses are placed. Since the resulting functions match for every x in R, composition is associative.

With that, (S, ◦) becomes a semigroup: the set of all real-to-real functions under composition. The payoff is that associativity makes longer compositions unambiguous without constant bookkeeping, setting the stage for exploring what semigroups can do next.