Magma

A magma — also called a groupoid or a binar — is a basic algebraic structure consisting of a set equipped with a single closed binary operation. It is the most primitive algebraic structure: it requires only closure, with no axioms of associativity, identity, or invertibility. Every group, semigroup, and monoid is a magma, but the converse does not hold.

The magma is the minimal structure from which all richer algebraic structures are built by adding axioms. Its very poverty makes it a useful laboratory for studying what happens when algebraic constraints are relaxed. The free magma on a set — the set of all binary trees with leaves labeled by elements of the set — captures the structure of non-associative operations in their most general form. In computer science, magmas appear in the study of binary operations on data types and in the algebraic specification of abstract data structures. In combinatorics, the enumeration of magmas on finite sets is a classic problem with connections to universal algebra and the theory of operads.