Infimum

A Infimum is the Set Element of a Partially Ordered Set that is GreaterThanOrEqualTo all Set Elements of some Subset of the set.

• AKA: Infima, Greatest Lower Bound, GLB.
  • …
• Counter-Example(s):
  • a Supremum.
• See: Numeric Interval, Lattice, Conceptual Graph.

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References

2009

• http://en.wikipedia.org/wiki/Infimum
  • In mathematics, particularly set theory, the infimum (plural infima) of a subset of some set is the greatest element (not necessarily in the subset) that is less than or equal to all elements of the subset. Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
  • Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found in that article.
  • In analysis the infimum or greatest lower bound of a subset [math]\displaystyle{ S }[/math] of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists (because [math]\displaystyle{ S }[/math] is not bounded below), then we define inf(S) = −∞. If [math]\displaystyle{ S }[/math] is empty, we define inf(S) = ∞ (see extended real number line).

2008

• (Corbett, 2008) ⇒ Dan R. Corbett. (2008). “Graph-based Representation and Reasoning for Ontologies.” In: Studies in Computational Intelligence, Springer. [http://dx.doi.org/10.1007/978-3-540-78293-3 10.1007/978-3-540-78293-3 doi:[http://dx.doi.org/10.1007/978-3-540-78293-3 10.1007/978-3-540-78293-3)
  • The greatest lower bound (GLB) of two CGs is the most general common specialization of the two conceptual graphs. Let G" be a specialization of G and G'. G" is the GLB of G and G' if, for any conceptual graph U where G ∨ G' = U, either G" ≥ U or G" = U.
  • The GLB of two graphs s and t is written as s | | t. Conversely, the most specific common generalization, known as the least upper bound (LUB), of two graphs is written s |_| t. Note that it is not always possible to find a unique GLB. In these instances, it is often the case that a greedy algorithm is used which picks the first G" which matches the constraints.

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