Definition of "maximal" in English

adjective

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

2. (mathematics, of a sequence) 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.

3. (mathematics, of an element of a poset) 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 #92;#123;#92;#123;2#92;#125;,#92;#123;1,2#92;#125;,#92;#123;2,3#92;#125;#92;#125; has two maximal elements: #92;#123;1,2#92;#125; and #92;#123;2,3#92;#125;

noun

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

2. (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).

3. (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.