Picking maxima and minima

Four commonly encountered functions are \(\max\), \(\min\), \(\mathrm{argmax}\), and \(\mathrm{argmin}\). Each one is fairly intuitive.

\(\max\) and \(\min\)

The \(\max\) function returns the largest element of a set.

Let \(S \mathrel{\mathop:}=\left \{ -5, 7, 23 \right \}\). Then \(\max(S) = \max(\left \{ -5, 7, 23 \right \}) = 23\). Some authors just write \(\max(-5, 7, 23)\) instead of \(\max(\left \{ -5, 7, 23 \right \})\).

In most cases, \(\max\) is used with numbers. But the function can be generalized to any structure that is a linear order.

Consider 2, the lattice with \(F < T\). Then \(\max(\left \{ F, T \right \}) = T\).

Note that the order must be a linear order. With weak partial orders that aren’t also linear orders, \(\max\) may not be defined for all cases.

Consider a case hierarchy with \(\text{Nom} \leq \text{Acc}\) and \(\text{Nom} \leq \text{Gen}\), but Acc and Gen are unordered with respect to each other. Then \(\max(\left \{ \text{Acc}, \text{Gen} \right \})\) is undefined.

The opposite of \(\max\) is \(\min\). It returns the smallest member of a set.

While \(\max(\left \{ -5, 7, 23 \right \} = 23\), \(\min(\left \{ -5, 7, 23 \right \} = -5\). And assuming \(F < T\), \(\min(\left \{ T, F \right \}) = F\).

argmax/argmin

will be added at a later point

softmax

will be added at a later point