# greatest lower bound

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Related to greatest lower bound: Least upper bound, Greatest lower bound property

## greatest lower bound

[′grād·əst ¦lō·ər ′bau̇nd]
(mathematics)
The greatest lower bound of a set of numbers S is the largest number among the lower bounds of S. Abbreviated glb. Also known as infimum (inf).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## greatest lower bound

(theory)
(glb, meet, infimum) The greatest lower bound of two elements, a and b is an element c such that c <= a and c <= b and if there is any other lower bound c' then c' <= c.

The greatest lower bound of a set S is the greatest element b such that for all s in S, b <= s. The glb of mutually comparable elements is their minimum but in the presence of incomparable elements, if the glb exists, it will be some other element less than all of them.

glb is the dual to least upper bound.

(In LaTeX "<=" is written as \sqsubseteq, the glb of two elements a and b is written as a \sqcap b and the glb of set S as \bigsqcap S).
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
References in periodicals archive ?
--The value associated to a formula of the form F = [inverted] Ax.F'(x) is computed by considering the greatest lower bound (glb) of the values associated to all the ground formulas F'(x/d), where d is any domain element.
Here we follow the same approach, but we have to generalize the notion of intersection of two models, written as "o", as their greatest lower bound in the lattice <A, [[is less than or equal to].sub.S]>.
For every ground atomic formula F and a family of models [{[M.sub.i]}.sub.i[element of]I], we define [o.sub.i[element of]I][M.sub.i](F) = [glb.sub.i[element of]I]{[M.sub.i](F)}, where glb is the greatest lower bound over the lattice <A, [[is less than or equal to].sub.S]>.
It is easy to see that <[IS.sub.P], ??> is a complete partial order, whose greatest lower bound coincides with the glb operation in the lattice A (suitable extended to interpretations).

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