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A binary operation on a set assigns to each ordered pair of elements a unique element of the set. Algebraic structures are sets equipped with operations satisfying specific axioms like closure, associativity, and identity properties. Understanding these foundational concepts is essential for studying groups, rings, and fields.
A binary operation on a set S is a rule that takes any two elements of S (in order) and produces another element. You've been using binary operations your entire mathematical life: addition takes two numbers and returns a number; multiplication does the same. What abstract algebra does is strip away the specific numbers and ask: what properties of the operation itself are doing the mathematical work?
The most fundamental property is closure: the result of the operation must stay inside the set. Addition is closed on the integers (adding two integers always gives an integer), but subtraction is not closed on the natural numbers (5 − 8 = −3, which leaves ℕ). Associativity says the order of grouping doesn't matter: (a ★ b) ★ c = a ★ (b ★ c). Addition and multiplication are associative; subtraction is not. Commutativity says order of inputs doesn't matter: a ★ b = b ★ a. Note that associativity and commutativity are independent properties — matrix multiplication is associative but not commutative.
Beyond closure and associativity, we look for an identity element: an element e such that e ★ a = a ★ e = a for all a. For addition, 0 is the identity; for multiplication, 1 is. An identity lets us define inverses: for each element a, an inverse a⁻¹ such that a ★ a⁻¹ = e. Every integer has an additive inverse (its negative), but not every integer has a multiplicative inverse within the integers (1/3 is not an integer). This is why groups, rings, and fields exist as distinct structures — each one specifies exactly which combination of these properties the operation satisfies.
Your prior work with equivalence relations connects directly here: when algebraic structures are studied, we routinely partition their elements into equivalence classes (cosets, for instance), and the binary operation must interact coherently with that partition structure. The concept of a well-defined operation — that the result doesn't depend on which representative of an equivalence class you pick — is where equivalence relations and binary operations meet. Getting this foundation right is what makes the more elaborate structures you'll encounter next (groups, homomorphisms, quotient structures) logically coherent.