Definition of "maximal"

Largest, greatest (in magnitude), highest, most.  examples

(mathematics, of a sequence) Larger than any previous term in the sequence.  examples

(mathematics) The element of a set with the greatest magnitude.  examples

(mathematics) 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).  examples

(logic) 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.  examples