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Associative**

Given a binary product \(\cdot:X\times X\to X\) on a set \(X\), one says that \(\cdot\) is associative if the order of calculation does not matter for repeated applications of \(\cdot\), i.e. \(\forall x,y,z\in X:x\cdot(y\cdot z)=(x\cdot y)\cdot z\). More generally if \(\circ\) is a partial function on \(X\), \(\circ\) is called associative if \(f\circ(g\circ h)=(f\circ g)\circ h\) whenever \(f,g,h\in X\) such that \(f\circ g,\:g\circ h,\:f\circ(g\circ h),\:(f\circ g)\circ h\) exist, for example composition of arrows in a category. Finally, given multiple sets and functions between them, \(\cdot:A\times A\to A\) and \(\star:A\times B\to B\), one says that \(\cdot,\star\) associate with each other if \(\forall \alpha,\beta\in A:\forall x\in B:\alpha\star(\beta\star x)=(\alpha\cdot\beta)\star x\), for example multiplication in the field and scalar multiplication for a vector space.