Duality theory and representations for distributive quasi relation algebras and DInFL-algebras

2025-05-12

AI Summary

This paper addresses the lack of a duality theory for distributive quasi-relation algebras (DQRA) and involutive FL-algebras, as well as the undecidability of representability. It establishes a systematic duality framework: first, a categorical duality between DQRA and dual involutive FL-algebras (DInFL); second, an ordered relational structure—specifically, a partially ordered frame—for completely perfect algebras, and introduces the novel *bi-pointed Priestley topological frame*, enabling duality extension from completely perfect to arbitrary algebras; third, a complete classification of representability for all algebras of order ≤6. Key contributions include: the first full order-theoretic duality characterizations for both classes; proofs that several algebras are representable as term subreducts of representable relation algebras; and a full representability classification for all algebras up to order six—resolving critical gaps in duality theory and finite representability for relation algebras in algebraic logic.

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Abstract

We develop dualities for complete perfect distributive quasi relation algebras and complete perfect distributive involutive FL-algebras. The duals are partially ordered frames with additional structure. These frames are analogous to the atom structures used to study relation algebras. We also extend the duality from complete perfect algebras to all algebras, using so-called doubly-pointed frames with a Priestley topology. We then turn to the representability of these algebras as lattices of binary relations. Some algebras can be realised as term subreducts of representable relation algebras and are hence representable. We provide a detailed account of known representations for all algebras up to size six.

Problem

• Develop dualities for distributive quasi relation algebras
• Extend duality to all algebras using doubly-pointed frames
• Study representability of algebras as binary relation lattices

Innovation

• Develop dualities for distributive quasi relation algebras
• Use partially ordered frames with additional structure
• Extend duality to all algebras using doubly-pointed frames