Compare

How two values of a partially ordered type compare: -1, 0 or 1, or NaN when they are incomparable.

Compare(a, b)How two values of a partially ordered type compare: -1, 0 or 1, or NaN when they are incomparable.

Typed signatures

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

Domain: Structures

Details

• The PartialOrder protocol's member: a type conforms by saying how it compares two of its values.
• Reflexive, antisymmetric and transitive; a LinearOrder never answers NaN.
• Under dominance, $[3,1,1,1]$ and $[2,2,2]$ are incomparable: NaN.
• Set partitions of one ground set under refinement (finer is lower): NaN unless one refines the other.
• Permutations of one length under the right weak order, as Sage's permutohedron_lequal (default side="right"): p is below q when every inversion of p, a pair of values a < b with b first, is one of q's. The left order is this one on inverses.

Examples

Basic #

In a linear order: $-1$, $0$ or $1$

live ↗

In$\mathrm{Compare}(3, 5)$Out$-1$

$[3,1]$ dominates $[2,2]$

live ↗

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

Properties #

$[3,1,1,1]$ and $[2,2,2]$ are incomparable under dominance: NaN

live ↗

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

The all-singletons partition refines every partition of its set

live ↗

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

Known value · Sage SetPartition refinement

$\{\{1,2\},\{3\}\}$ and $\{\{1\},\{2,3\}\}$ neither refines the other: NaN

live ↗

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

Known value · Sage SetPartition refinement

Right weak order, as Sage's permutohedron_lequal: the inversion pairs of values $\{(2,3)\}$ lie inside $\{(1,3),(2,3)\}$

live ↗

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

Known value · Sage permutohedron_lequal

$213$ and $132$ invert different value pairs: NaN

live ↗

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

Known value · Sage permutohedron_lequal

About

Examples	7

See also: Min, Max, GreatestLowerBound