[Paper Review] Distributivity in skew lattices

Michael Kinyon, Jonathan Leech+1 more|arXiv (Cornell University)|Jun 24, 2013

Advanced Algebra and Logic17 references4 citations

TL;DR

This paper investigates distributivity in skew lattices, focusing on linearly distributive skew lattices—those whose totally preordered subalgebras are distributive. It characterizes linear distributivity via the natural partial order on 𝒟-classes and introduces a second characterization using strictly categorical skew lattices, establishing that distributivity is equivalent to both quasi-distributivity and linear distributivity in symmetric skew lattices.

ABSTRACT

Distributive skew lattices satisfying $x\wedge (y\vee z)\wedge x = (x\wedge y\wedge x) \vee (x\wedge z\wedge x)$ and its dual are studied, along with the larger class of linearly distributive skew lattices, whose totally preordered subalgebras are distributive. Linear distributivity is characterized in terms of the behavior of the natural partial order between comparable $\DD$-classes. This leads to a second characterization in terms of strictly categorical skew lattices. Criteria are given for both types of skew lattices to be distributive.

Motivation & Objective

• To characterize linear distributivity in skew lattices using the natural partial order between 𝒟-classes.
• To establish a second characterization of linear distributivity through strictly categorical skew lattices.
• To determine necessary and sufficient conditions for skew lattices to be distributive, particularly in symmetric and quasi-distributive cases.
• To clarify the hierarchy among distributivity concepts: biconditional, relative, linear, and full distributivity.
• To show that biconditionally distributive and symmetric linearly distributive skew lattices are relatively distributive, forming varieties.

Proposed method

• Characterizes linear distributivity via the behavior of the natural partial order on 𝒟-classes in skew lattices.
• Introduces a second characterization using strictly categorical skew lattices, linking algebraic structure to order-theoretic properties.
• Applies equational reasoning and Prover9 to verify identities, particularly for upper and lower symmetric skew lattices.
• Uses absorption and regularity identities (e.g., x ∨ (y ∧ z) ∧ x = (x ∨ y ∧ x) ∨ (x ∨ z ∧ x)) to derive distributive laws.
• Analyzes subalgebras and embeddings to distinguish between relative, linear, and full distributivity.
• Establishes that biconditional distributivity implies relative distributivity, and that both form varieties.

Experimental results

Research questions

• RQ1 Under what conditions does the identity x ∧ (y ∨ z) ∧ x = (x ∧ y ∧ x) ∨ (x ∧ z ∧ x) hold in skew lattices?
• RQ2 How does the natural partial order on 𝒟-classes relate to linear distributivity in skew lattices?
• RQ3 What characterizes linearly distributive skew lattices in terms of categorical structure?
• RQ4 When is a quasi-distributive skew lattice fully distributive?
• RQ5 What is the relationship between biconditional, relative, and linear distributivity in skew lattices?

Key findings

• Linear distributivity in skew lattices is characterized by the behavior of the natural partial order between 𝒟-classes.
• A second characterization of linear distributivity is provided via strictly categorical skew lattices.
• A symmetric skew lattice is distributive if and only if it is both quasi-distributive and linearly distributive.
• A simply cancellative skew lattice is distributive if and only if it is linearly distributive.
• Biconditionally distributive skew lattices and symmetric linearly distributive skew lattices are relatively distributive, forming varieties.
• Relative distributivity is strictly stronger than linear distributivity, and biconditional distributivity is strictly stronger than relative distributivity.