Representability for Distributive Quasi Relation Algebras via Nested Sums

Andrew Craig; Wilmari Morton; Claudette Robinson

doi:10.1007/s11083-026-09732-5

Representability for Distributive Quasi Relation Algebras via Nested Sums

Andrew Craig
, Wilmari Morton
, Claudette Robinson

Mathematics and Applied Mathematics

National Institute for Theoretical and Computational Sciences (NITheCS)
University of the Witwatersrand
University of Johannesburg

Research output: Contribution to journal › Article › peer-review

Abstract

We extend the work of Galatos (2004) on nested sums, originally called generalised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

Original language	English
Article number	19
Journal	Order
Volume	43
Issue number	1
DOIs	https://doi.org/10.1007/s11083-026-09732-5
Publication status	Published - Apr 2026

Keywords

Generalised ordinal sum
Nested sum
Quasi relation algebra
Representability
Sugihara chain

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s11083-026-09732-5

Cite this

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abstract = "We extend the work of Galatos (2004) on nested sums, originally called generalised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.",

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N2 - We extend the work of Galatos (2004) on nested sums, originally called generalised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

AB - We extend the work of Galatos (2004) on nested sums, originally called generalised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

KW - Generalised ordinal sum

KW - Nested sum

KW - Quasi relation algebra

KW - Representability

KW - Sugihara chain

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JO - Order

JF - Order

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ER -

Representability for Distributive Quasi Relation Algebras via Nested Sums

Abstract

Keywords

ASJC Scopus subject areas

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New Findings from University of Johannesburg in the Area of Mathematics Reported (Representability for Distributive Quasi Relation Algebras Via Nested Sums)

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Representability for Distributive Quasi Relation Algebras via Nested Sums

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Keywords

ASJC Scopus subject areas

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New Findings from University of Johannesburg in the Area of Mathematics Reported (Representability for Distributive Quasi Relation Algebras Via Nested Sums)

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