Algebra 2 | Semigroups [dark version]

TL;DR

A binary operation on a set A is a function F: A×A → A, so combining any two elements must produce another element of A.

Briefing Cornell Notes

Briefing

Semigroups start with a simple idea: take a set and a rule for combining any two elements, then require one key law so the order of parentheses doesn’t change the outcome. That “parentheses don’t matter” condition is associativity, and it turns an otherwise messy binary operation into a structured algebraic object. The payoff is practical: once associativity holds, expressions can be written without constantly tracking grouping.

The foundation is a binary operation. Given a set A, a binary operation is a function F from A×A back to A, meaning every pair (a,b) in A produces a result still inside A. This is the closure requirement: combining two elements of A must never leave the set. When the operation is restricted to a subset, closure must be checked again for that subset, because not every pair in the subset is guaranteed to land back in it.

For finite sets, the operation can be encoded in an operation table. Each entry specifies the result of combining the element from the row with the element from the column. The table also makes clear why commutativity is not automatic: in general, a∘b can differ from b∘a. Likewise, without extra properties, changing parentheses can change the result. Using the table, (1∘2)∘3 can yield a different element than 1∘(2∘3), since the intermediate combination changes what gets fed into the next step.

Associativity is exactly the fix for that parentheses problem. A semigroup is defined as a pair (S,∘) consisting of a set S and an associative binary operation ∘. Associativity means that for all a,b,c in S, the equation (a∘b)∘c = a∘(b∘c) holds. When this law is satisfied, parentheses can be omitted in longer products as long as the overall left-to-right order of elements is preserved.

A concrete example comes from functions. Let S be the set of functions from R to R. The binary operation is function composition: f∘g means composing two functions to produce a new function. Composition is closed here because composing two functions R→R again yields a function R→R. More importantly, composition is associative: for any three functions F1, F2, F3 and any input x in R, both (F1∘F2)∘F3 and F1∘(F2∘F3) produce the same output when evaluated at x. Since the resulting functions agree on every input, the two ways of parenthesizing coincide.

With that, the semigroup concept is pinned down: it’s not about having inverses or identities like groups, but about having a closed binary operation that behaves consistently under reassociation. That consistency is what makes semigroups a useful stepping stone for later algebraic structures.

Cornell Notes

A semigroup is a set S equipped with a binary operation ∘ such that combining any two elements stays inside S (closure) and the operation is associative. Associativity requires that for all a, b, c in S, (a∘b)∘c equals a∘(b∘c), so parentheses can be dropped without changing the result. Binary operations can be represented by operation tables, where commutativity and associativity are not guaranteed. A key example uses functions from R to R with the operation of composition: composing functions is closed and associative, because both parenthesizations give the same output for every input x. This makes (S,∘) a semigroup and shows why associativity is the central rule behind simplifying algebraic expressions.

What exactly qualifies as a binary operation on a set A?

A binary operation on A is a rule F: A×A → A. For every pair (a,b) with a,b in A, the result F(a,b) must be an element of A again. In practice, this is often written as a∘b (or sometimes ab or a+b), but the defining requirement is that the output always lands back in A.

Why does closure matter when restricting a binary operation to a subset?

Closure guarantees that combining two elements produces an element still inside the set. If an operation is restricted to a subset T⊆A, closure must be rechecked: for all t1,t2 in T, the result t1∘t2 must lie in T. Without that check, the restricted rule might send pairs outside the subset, so it would no longer be a binary operation on T.

How do operation tables show that commutativity and associativity are not automatic?

An operation table lists the result of a∘b for each row/column pair. If the entry for (a,b) differs from the entry for (b,a), then a∘b ≠ b∘a, so commutativity fails. Similarly, comparing (a∘b)∘c with a∘(b∘c) can yield different table-driven outcomes because the intermediate result changes depending on parentheses.

What is the precise associativity condition used to define a semigroup?

Associativity requires that for all a,b,c in S, (a∘b)∘c = a∘(b∘c). This equality must hold universally across the set, not just for selected elements. Once it holds, parentheses can be omitted in longer expressions without altering the final result, as long as the element order is kept.

Why is function composition an associative binary operation on functions R→R?

Take three functions F1, F2, F3 in S and an input x in R. For G = (F1∘F2)∘F3, evaluating at x gives F1(F2(F3(x))). For H = F1∘(F2∘F3), evaluating at x also gives F1(F2(F3(x))). Since both parenthesizations produce the same output for every x, the resulting functions are identical, so composition is associative.

Review Questions

Given a set A and a proposed rule ∘, what two properties must be verified to claim (A,∘) is a semigroup?
Using an operation table, how would you test whether parentheses matter for a∘b∘c?
Explain why associativity allows dropping parentheses but does not allow changing the overall order of elements.

Key Points

1
A binary operation on a set A is a function F: A×A → A, so combining any two elements must produce another element of A.
2
Closure is built into the definition of a binary operation, but must be rechecked when restricting the operation to a subset.
3
Operation tables encode the result of a∘b for each pair and make it easy to see when commutativity fails.
4
Without associativity, changing parentheses can change the result, because intermediate combinations differ.
5
A semigroup is a pair (S,∘) where ∘ is associative: (a∘b)∘c = a∘(b∘c) for all a,b,c in S.
6
Associativity is what permits writing longer expressions without parentheses while preserving the left-to-right order.
7
Function composition on functions from R to R is closed and associative, providing a standard semigroup example.

Highlights

A semigroup is defined by one central law: associativity, which guarantees (a∘b)∘c = a∘(b∘c) for all elements.

Closure is the non-negotiable requirement that combining two elements never leaves the set.

Operation tables reveal immediately that commutativity and associativity are not guaranteed by default.

Function composition on R→R functions is associative because both parenthesizations yield the same value for every input x.

Topics

Semigroups
Binary Operations
Associativity
Function Composition
Operation Tables