A binary operation on a set combines any two elements to produce another element. Algebraic structures are sets with operations satisfying specific axioms. Understanding closure, associativity, identity, and inverses distinguishes different algebraic structures and forms the foundation for groups, rings, and fields.
A binary operation on a set S is a rule that takes any two elements of S and produces a third element of S. Ordinary addition on the integers is a binary operation: take any two integers, add them, get an integer. But subtraction on the natural numbers ℕ = {0, 1, 2, ...} is not a binary operation on ℕ, because 3 − 5 = −2 is not in ℕ. This first property — that the output stays in the same set — is called closure. It is not automatic; you must always check it for the specific set and operation.
Once you have a closed binary operation, you can ask which additional properties it satisfies. Associativity means (a ★ b) ★ c = a ★ (b ★ c) for all elements: you can regroup without changing the result. Addition and multiplication are associative; subtraction is not ((5 − 3) − 1 ≠ 5 − (3 − 1)). An identity element e satisfies e ★ a = a ★ e = a for all a — it acts as a "do nothing" element. Zero is the identity for addition, one is the identity for multiplication. An inverse of a (when it exists) is an element a' such that a ★ a' = a' ★ a = e: the element that "undoes" a. Every integer has an additive inverse (its negative), but only ±1 have multiplicative inverses in the integers.
These four properties — closure, associativity, identity, inverses — are the ingredients of a group, the central object of abstract algebra. But not every operation has all four. A magma has only closure. A semigroup adds associativity. A monoid adds an identity. A group adds inverses. This hierarchy helps you classify any structure you encounter: identify which axioms hold and you know exactly what tools are available to you.
The power of this framework is that it abstracts across wildly different examples: the integers under addition, the symmetries of a triangle, nonzero rationals under multiplication, and 2×2 invertible matrices all satisfy the group axioms. Any theorem proved using only those axioms applies to all of them simultaneously. Your prerequisite work on equivalence relations will resurface here, since cosets partition a group into equivalent classes and lead directly to quotient structures.