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An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema. Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds.
A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real ...
The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields ...
A complete lattice is a lattice in which every subset of elements of L has an infimum and supremum; this generalizes the analogous properties of the real numbers. An order-embedding is a function that maps distinct elements of S to distinct elements of L such that each pair of elements in S has the same ordering in L as they do in S.
Every subset of a complete Boolean algebra has a supremum, by definition; it follows that every subset also has an infimum (greatest lower bound). For a complete boolean algebra, both infinite distributive laws hold if and only if it is isomorphic to the powerset of some set. [citation needed]
The supremum is given by the union and the infimum by the intersection of subsets. The non-negative integers ordered by divisibility. The least element of this lattice is the number 1 since it divides any other number. Perhaps surprisingly, the greatest element is 0, because it can be divided by any other number.
An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.