[Paper Review] Examples, counterexamples, and structure in bounded width algebras

Zarathustra Brady|arXiv (Cornell University)|Sep 12, 2019

Advanced Algebra and Logic18 references4 citations

TL;DR

This paper investigates minimal bounded width algebras—those with bounded relational width but no proper reducts retaining this property—by introducing a pseudovariety structure via a ternary operation. It classifies such algebras of size ≤3 and establishes a structure theorem for those without majority subalgebras, showing they form a pseudovariety with a commutative binary operation, while also classifying minimal clones with Taylor terms as a byproduct.

ABSTRACT

We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We classify minimal bounded width algebras which have size at most three, and prove a structure theorem for minimal bounded width algebras which have no majority subalgebra, which form a pseudovariety with a commutative binary operation. As a byproduct of our results, we also classify minimal clones which have a Taylor term.

Motivation & Objective

• To classify minimal bounded width algebras of size at most three.
• To characterize minimal bounded width algebras without majority subalgebras via a pseudovariety with a commutative binary operation.
• To establish a pseudovariety structure for minimal bounded width algebras using a basic ternary operation.
• To provide a classification of minimal clones with a Taylor term as a byproduct of the main results.
• To investigate structural conjectures on term equivalence, subalgebra generation, and connectivity in minimal bounded width algebras.

Proposed method

• Uses $pq$ instances and cycle consistency to derive a direct proof for the existence of weak near-unanimity terms in bounded width algebras.
• Introduces a new characterization of bounded width via existence of idempotent terms $f$ and $g$ satisfying $g(x,x,y) o f(x,y)$ and $f(f(x,y),f(y,x)) o f(x,y)$.
• Applies an iteration argument on terms to construct an infinite family of weak near-unanimity terms of high arity.
• Employs the concept of subalgebras generated by pairs to analyze structural properties, particularly in the absence of majority subalgebras.
• Uses the invariant $ ext{Inv}_2( ext{A})$, the collection of subalgebras of $ ext{A} imes ext{A}$, to explore term equivalence and structural recovery.
• Leverages known equivalences from prior work (e.g., [13], [18], [21]) to reduce the problem to Mal’cev conditions and congruence properties.

Experimental results

Research questions

• RQ1 What is the complete classification of minimal bounded width algebras of size at most three?
• RQ2 How can minimal bounded width algebras without majority subalgebras be structurally characterized?
• RQ3 To what extent does $ ext{Inv}_2( ext{A})$ determine a minimal bounded width algebra up to term equivalence?
• RQ4 Do minimal bounded width algebras satisfy the conjectured identities involving terms $w$ and $s$ resembling majority and semilattice operations?
• RQ5 Can the structure of minimal bounded width algebras be built from smaller subalgebras via specific term operations?

Key findings

• All minimal bounded width algebras of size at most three are classified, with explicit descriptions of their term operations and subalgebra lattices.
• Minimal bounded width algebras without majority subalgebras form a pseudovariety closed under subalgebras and products, equipped with a commutative binary operation.
• An infinite family of weak near-unanimity terms is constructed for any bounded width algebra, with arity exceeding $2 imes ext{lcm}igrace{1,2, ext{...},| ext{A}|-1igrace}$, satisfying strong stability conditions.
• The existence of a term $t(x,y)$ satisfying $t(x,t(x,y)) o t(x,y)$ is proven for all bounded width algebras, supporting the conjectured semilattice-like behavior.
• The classification of minimal clones with a Taylor term is completed as a byproduct, showing they coincide with minimal bounded width algebras without affine quotients.
• Conjecture 4 is supported: minimal bounded width algebras are determined up to term equivalence by $ ext{Inv}_2( ext{A})$, the set of subalgebras of $ ext{A} imes ext{A}$.