[Paper Review] Relation identities equivalent to congruence modularity

Paolo Lipparini|arXiv (Cornell University)|Apr 18, 2017

Advanced Algebra and Logic10 references3 citations

TL;DR

This paper presents new relation identities involving reflexive and admissible relations that are equivalent to congruence modularity in universal algebra. By leveraging directed Gumm terms and relational compositions, the author establishes that identities such as $\Theta(S \circ S) \subseteq (\Theta S)^*$ characterize congruence modularity, offering a novel algebraic characterization beyond traditional term-based definitions.

ABSTRACT

We present some identities dealing with reflexive and admissible relations and which, through a variety, are equivalent to congruence modularity.

Motivation & Objective

• To identify new relational identities that are equivalent to congruence modularity in varieties of algebras.
• To extend the characterization of congruence modularity beyond term-based identities to include identities involving reflexive and admissible relations.
• To clarify the relationship between relational identities and the existence of directed Gumm terms or Day terms in congruence modular varieties.
• To provide a framework where tolerance and congruence identities can be systematically analyzed using relational composition and closure operations.

Proposed method

• The paper introduces relational notations: juxtaposition for intersection, $\circ$ for composition, $R^{\smallsmile}$ for converse, $R^*$ for transitive closure, and $\overline{R}$ for the smallest reflexive and admissible relation containing $R$.
• It defines $\Theta_R$ as the smallest tolerance containing a reflexive and admissible relation $R$, enabling the use of tolerances as variables in identities.
• The key method involves proving that certain relational inclusions—such as $\Theta(S \circ S) \subseteq (\Theta S)^*$—hold in a variety if and only if the variety is congruence modular.
• It establishes equivalence between these relational identities and the existence of directed Gumm terms, using a chain of identities derived from the terms' axioms.
• The paper uses the concept of $S + T = \bigcup_{m \in \mathbb{N}} S \circ_m T$ to model joins of relations and relates this to congruence lattice operations.
• It proves that the identity $\Theta(S \circ S^\smallsmile) \subseteq \Theta S \circ_{k-1} \Theta S^\smallsmile$ characterizes the existence of $k+1$ Day terms, linking relational identities to the number of terms in a Day term sequence.

Experimental results

Research questions

• RQ1 Which identities involving reflexive and admissible relations are equivalent to congruence modularity in a variety?
• RQ2 How do relational compositions and closures relate to the existence of directed Gumm terms?
• RQ3 Can the number of Day terms in a congruence modular variety be characterized by a relational identity?
• RQ4 What happens when small modifications are made to the identities—do they imply stronger properties like distributivity or m-permutability?
• RQ5 To what extent can tolerance-based identities replace term-based characterizations in universal algebra?

Key findings

• The identity $\Theta(S \circ S) \subseteq (\Theta S)^*$ holds in a variety if and only if the variety is congruence modular.
• The identity $\Theta(S \circ S^\smallsmile) \subseteq (\Theta S \circ \Theta S^\smallsmile)^*$ is equivalent to congruence modularity.
• The identity $\Theta(S \circ T)^* \subseteq \Theta(\overline{S \cup T}) \circ (\Theta S \circ \Theta T)^*$ characterizes congruence modularity.
• The identity $\Theta(S \circ S^\smallsmile) \subseteq \Theta S \circ_{k-1} \Theta S^\smallsmile$ holds if and only if the variety has $k+1$ Day terms.
• A variation of identity (1.1), $\Theta(S \circ S) \subseteq (\Theta S^\smallsmile)^*$, is strictly stronger than modularity and implies m-permutability for some m.
• The identity $\Theta(S \circ T^\smallsmile) \subseteq (\Theta S \circ \Theta T^\smallsmile)^*$ is equivalent to congruence distributivity, not modularity.