XYZ inequality¶
A correlation inequality constraining relative ordering probabilities for three incomparable elements in a finite partially ordered set.
Core Idea¶
The Fishburn–Shepp XYZ inequality relates probabilities such as P(x before y) under a uniformly random linear extension and asserts that not all three cyclic comparisons can be too strongly biased in the same direction. Linear extensions supply a probability space, involutions or correlation arguments pair orders and the partial-order constraints bound the product or combination of cyclic precedence probabilities. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
XYZ inequality belongs to order theory and is useful where the analyst can specify the typed order theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite poset, three incomparable elements, uniform linear-extension measure, exact precedence probabilities and stated XYZ inequality and equality conditions are explicit. The scope is broad within that domain but bounded by the need for the finite poset, three incomparable elements, uniform linear-extension measure, exact precedence probabilities and stated XYZ inequality and equality conditions are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite poset, three incomparable elements, uniform linear-extension measure, exact precedence probabilities and stated XYZ inequality and equality conditions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name XYZ inequality can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to XYZ inequality. XYZ inequality compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed order theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite poset, three incomparable elements, uniform linear-extension measure, exact precedence probabilities and stated XYZ inequality and equality conditions are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of order theory because they reuse the typed order theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Linear extensions supply a probability space, involutions or correlation arguments pair orders and the partial-order constraints bound the product or combination of cyclic precedence probabilities., and type the carrier, state every parameter and convention in the definition, test that the finite poset, three incomparable elements, uniform linear-extension measure, exact precedence probabilities and stated XYZ inequality and equality conditions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction XYZ inequality Domain-specific
Parents (1) — more general patterns this builds on
-
XYZ inequality is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- XYZ inequality → Constraint
Neighborhood in Abstraction Space¶
XYZ inequality sits in a crowded region of the domain-specific corpus (11th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Order, Lattices & Set Relations (36 abstractions)
Nearest neighbors
- Join and meet — 0.93
- Complete lattice — 0.93
- Interval order — 0.93
- Partially ordered set — 0.93
- Sperner property of a partially ordered set — 0.93
Computed from structural-signature embeddings · 2026-09-08