adjective Largest, greatest (in magnitude), highest, most.
adjective Larger than any previous term in the sequence.
In the sequence (1, 2, 10, 5, 12, 6), the fifth term, f(5) = 12, is a maximal term, as each of the first 4 terms are smaller than 12.
adjective Such that no other element is greater (with respect to the given partial order).
With respect to the ordering induced by set-theoretic inclusion, the set {{2},{1,2},{2,3}} has two maximal elements: {1,2} and {2,3}
noun The element of a set with the greatest magnitude.
noun Said of an ideal of a ring or a filter of a lattice: that it is as large as it can be without being trivial (improper).
noun Said of a set of well-formed formulas: that it is as large as it can be without being inconsistent; i.e. that for any well-formed formula φ, the set contains either φ or ~φ.