GreatestLowerBound

The meet of two values in a lattice: the greatest value at or below both.

GreatestLowerBound(a, b)The meet of two values in a lattice: the greatest value at or below both.

Typed signatures

Type	Package	Applies
(any, any) -> unknown	structures	always

Domain: Structures

Details

• The Lattice protocol's member, and what Min folds when its arguments' type is a lattice.
• In a linear order it is the smaller of the two; under dominance it takes the partial sums' pointwise minimum.
• Set partitions: the common refinement, the non-empty pairwise block intersections.
• Permutations (right weak order): the complement of the transitive closure of the union of the complements of the inversion sets.

Examples

Basic #

In a linear order, the smaller of the two

live ↗

In$\mathrm{GreatestLowerBound}(3, 5)$Out$3$

Scope #

Under dominance, the meet of two incomparable partitions

live ↗

In$\mathrm{GreatestLowerBound}({\operatorname{IntegerPartition}}([3, 1, 1, 1]), {\operatorname{IntegerPartition}}([2, 2, 2]))$Out${\operatorname{IntegerPartition}}([2, 2, 1, 1])$

Under refinement, the meet is the common refinement: the non-empty pairwise block intersections

live ↗

In$\mathrm{GreatestLowerBound}({\operatorname{SetPartition}}([[1, 2], [3]]), {\operatorname{SetPartition}}([[1], [2, 3]]))$Out${\operatorname{SetPartition}}([[1], [2], [3]])$

Known value · Sage SetPartition.inf

Under the right weak order the meet of the two 3-cycles is the identity

live ↗

In$\mathrm{GreatestLowerBound}({\operatorname{Permutation}}([2, 3, 1]), {\operatorname{Permutation}}([3, 1, 2]))$Out${\operatorname{Permutation}}([1, 2, 3])$

Known value · Sage LatticePoset of permutohedron_lequal meet

About

Examples	4

See also: LeastUpperBound, Min, Compare