 # Maximal $LaTeX: \max \{ 5x + y: (x, y) in \mathbb{Z}^2, \, x \ge 0, \, y \ge 0, 3x + 2y \le 3\}.$
The maximum value is $LaTeX: 5$, with $LaTeX: (x, y) = (1, 0).$ Define the partial ordering to be the ordinary greater-than-or-equal-to, so that $LaTeX: (x',y') > (x, y)$ means $LaTeX: x' \ge x,$ $LaTeX: y' \ge y,$ and either $LaTeX: x' > x$ or $LaTeX: y' > y$. In words, $LaTeX: (x', y') > (x, y)$ means we can put at least one more item into the knapsack, in which case we would increase our return. When we cannot do so, $LaTeX: (x, y)$ is a maximal element. In particular, $LaTeX: (0,1)$ is a maximal element, and its value is $LaTeX: 1$, which is not the maximum.
Another definition pertains to a subset with respect to some property such that no proper superset has that property. In the 0-1 knapsack problem with $LaTeX: n$ variables, we define the subset of items taken (for any solution) as $LaTeX: J=\{j: x_j=1\}.$ Then, we define $LaTeX: J$ as maximal if no item can be added without violating the limit, i.e., $LaTeX: \textstyle a_k > b - \sum_{j\in J} a_j$ for all $LaTeX: k$ not in $LaTeX: J.$