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A lattice-ordered set is a poset in which each two-element subset has an infimum, denoted , and a supremum, denoted . There is a natural relationship between lattice-ordered sets and lattices. In fact, a lattice is obtained from a lattice-ordered poset by defining and for any . Also, from a lattice , one may obtain a lattice-ordered set by setting in if and only if . One obtains the same lattice-ordered set from the given lattice by setting in if and only if . (In other words, one may prove that for any lattice, , and for any two members and of , if and only if .)
Lattice-ordered sets abound in mathematics and its applications, and many authors do not distinguish between them and lattices. From a universal algebraist's point of view, however, a lattice is different from a lattice-ordered set because lattices are algebraic structures that form an equational class or variety, but lattice-ordered sets are not algebraic structures, and therefore do not form a variety.
A lattice-ordered set is bounded provided that it is a bounded poset, i.e., if it has an upper bound and a lower bound. For a bounded lattice-ordered set, the upper bound is frequently denoted 1 and the lower bound is frequently denoted 0. Given an element of a bounded lattice-ordered set , we say that is complemented in if there exists an element such that and .
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تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
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أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
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قسم شؤون المعارف ينظم دورة عن آليات عمل الفهارس الفنية للموسوعات والكتب لملاكاته
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