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lowest upper bound (Definition)

Let $ S$ be a set with a partial ordering $ \leq$, and let $ T$ be a subset of $ S$. A lowest upper bound, or supremum, of $ T$ is an upper bound $ x$ of $ T$ with the property that $ x \leq y$ for every upper bound $ y$ of $ T$. The lowest upper bound of $ T$, when it exists, is denoted $ \operatorname{sup}(T)$.

A lowest upper bound of $ T$, when it exists, is unique.

Greatest lower bound is defined similarly: a greatest lower bound, or infimum, of $ T$ is a lower bound $ x$ of $ T$ with the property that $ x \geq y$ for every lower bound $ y$ of $ T$. The greatest lower bound of $ T$, when it exists, is denoted $ \operatorname{inf}(T)$.

If $ A = \{a_1,a_2,\ldots,a_n\}$ is a finite set, then the supremum of $ A$ is simply $ \max(A)$, and the infimum of $ A$ is equal to $ \min(A)$.



"lowest upper bound" is owned by djao.
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Also defines:  least upper bound, greatest lower bound, supremum, infimum
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Cross-references: finite set, lower bound, property, upper bound, subset, partial ordering
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This is version 8 of lowest upper bound, born on 2001-10-21, modified 2007-02-15.
Object id is 452, canonical name is LowestUpperBound.
Accessed 18860 times total.

Classification:
AMS MSC: 06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order)
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