# maximal element

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## maximal element

[′mak·sə·məl ′el·ə·mənt]
(mathematics)
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In Section 3, we shall consider the ordered set [C.sub.[phi]] with order [subset] and investigate under what conditions the ordered set [C.sub.[phi]] has a maximal element, a minimal element, the smallest element, and the largest element.
The consideration that an element [x.sub.0] [member of] X is a (weak) Pareto optimal solution to problem (1) if and only if [x.sub.0] [member of] X is a maximal element for the preorder [mathematical expression not reproducible], respectively) and the observation that the function u = ([u.sub.1], ..., [u.sub.m]): X [??] [R.sup.m] is a (Richter-Peleg) multiutility representation of the preorder [mathematical expression not reproducible], respectively) allow us to present various results concerning the existence of solutions to the multiobjective optimization problem, also in the classical case when the design space is a compact topological space.
If every total ordered subset in E has an upper bound in E, then there is a maximal element in E.
It follows now from Zorn's Lemma that Q has a maximal element, denoted by [K.sub.[infinity]]: We claim that [K.sub.[infinity]] is a singleton.
([alpha]) If [S.sup.-] is transfer open-valued, then either (i) T has a fixed point, or (ii) S has a maximal element in K.
Note that the maximal chain from 0 to a maximal element of J maybe not unique, and the expression [[union].sup.r.sub.i=l][[bar.L].sub.i] may be not unique in general.
As [{v(P([a.sub.[rho]]))}.sub.[rho] < [kappa]] is cofinal in C, v(P(l)) would be a maximal element of C: contradiction.
Let [max.sub.A] be the maximal element in A and let [min.sub.A] be the minimal element in A.
For an integral neutrosophic lattice N also {0} is the minimal element and {1} is the maximal element of N.
F is a soft fuzzy T'-ultrafilter if F is a maximal element in the set of soft fuzzy T'-prefilters ordered by the inclusion relation.
Proposition 3.16: Let [mu] [member of]N(R) be non-constant such that it is a maximal element of (N(R), [subset or equal to]) then it takes only two values {0,1}.
In a traversal from a minimal element to a maximal element in partial order, ineffective paths (those under the lower bound of the similarity) are ignored, i.e., {[b.sub.3],[w.sub.4]}-{[b.sub.4],[w.sub.4]}, {[b.sub.2],[w.sub.4]}{[b.sub.4],[w.sub.4]}, {[b.sub.4],[w.sub.1]}-{[b.sub.4],[w.sub.4]}, and {[b.sub.4],[w.sub.2]}-{[b.sub.4],[w.sub.4]}, which should be are removed from the Hasse diagram of Fig.

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