LeastUpperBound

The join of two values in a lattice: the least value at or above both.

LeastUpperBound(a, b)The join of two values in a lattice: the least value at or above both.

Typed signatures

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

Domain: Structures

Details

• The Lattice protocol's member, and what Max folds when its arguments' type is a lattice.
• Set partitions under refinement: the finest common coarsening.
• Permutations (right weak order): the transitive closure of the union of the inversion sets.

Examples

Basic #

In a linear order, the larger of the two

live ↗

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

Scope #

Under dominance, the join of two incomparable partitions

live ↗

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

Under refinement, the join is the finest common coarsening: blocks sharing an element are glued

live ↗

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

Known value · Sage SetPartition.sup

Under the right weak order the join takes the transitive closure of the union of the inversion pairs

live ↗

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

Known value · Sage LatticePoset of permutohedron_lequal join

About

Examples	4

See also: GreatestLowerBound, Max, Compare