Magma

A magma is a set equipped with a binary operation that assigns an element of the set to every ordered pair of its elements. No algebraic laws are required.

Set: A collection of distinct objects treated as a single mathematical object. A magma's elements form the set on which its operation acts.
Semigroup: A set with an associative binary operation. Adding associativity to a magma produces a semigroup.
Cayley table: A table displaying the results of a binary operation on every pair of elements in a finite set. A finite magma's operation can be specified completely by its Cayley table.
Universal algebra: The study of algebraic structures through operations and identities shared across different systems. Magmas exemplify structures described by operations with few or no identities.
Binary operation: A rule assigning an output to each ordered pair of inputs from a specified domain. The operation combines two magma elements and must return another element of the set.
Monoid: A set with an associative binary operation and an identity element. A monoid adds associativity and an identity to the magma framework.
Nonassociative algebra: An algebraic system whose multiplication does not necessarily satisfy associativity. Magma operations can fail associativity, though a magma need not have the extra structure of an algebra.
Term algebra: An algebra whose elements are formal terms built from variables and operation symbols. Terms formed with one binary symbol provide a concrete free magma.
Closure (mathematics): A property stating that applying an operation to elements of a set yields an element of that set. Closure is the defining requirement that keeps a magma's operation inside its underlying set.
Quasigroup: A set with a binary operation for which each equation formed by fixing one input has a unique solution. A quasigroup strengthens a magma by requiring unique division.

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Inverse elementNarrower topic: This is the minimal algebraic setting in which the inverse question can be posed.