Associativity equation

The associativity equation or associativity functional equation is the functional equation

${\displaystyle F(F(x,y),z)=F(x,F(y,z))}$

for a function ${\displaystyle F\colon X\times X\to X}$. It characterizes those binary operations ${\displaystyle F}$ on a set ${\displaystyle X}$ that are associative in the usual algebraic sense, and therefore underlies the study of semigroups and many kinds of aggregation operators. When additional regularity conditions (such as continuity and monotonicity) are imposed, the equation has a rich and fairly explicit theory of solutions.