Relation algebras as expanded FL-algebras

Galatos, Nikolaos; Jipsen, Peter

doi:10.1007/s00012-012-0215-y

Cited by 4 publications

(3 citation statements)

References 20 publications

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“…The smallest Boolean cyclic involutive GBI-algebra that fails the converse identity has 8 elements and was originally found in the context of residuated lattices with a De Morgan operation [4]. This algebra has atoms 1, a, b and satisfies aa = a, bb = a ∨ b and ab = = ba.…”

Section: Coupled Semiringsmentioning

confidence: 99%

Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

Jipsen

2017

Relational and Algebraic Methods in Computer Science

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This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.

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Section: Coupled Semiringsmentioning

confidence: 99%

Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

Jipsen

2017

Relational and Algebraic Methods in Computer Science

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“…Another subvariety of FL -algebras is the variety of skew relation algebras [ 6 ], defined in this setting as Boolean involutive FL -algebras. As mentioned before, the variety of relation algebras is obtained by adding the identity where ([ 6 ], Cor. 29).…”

Section: Introductionmentioning

confidence: 99%

“…The setting of FL is well suited to studying weaker versions of this logic that omit some rules like contraction and/or weakening. It is also worth noting that classical BI-algebras coincide with commutative skew relation algebras (defined in [ 6 ]).…”

Section: Introductionmentioning

confidence: 99%

Weakening Relation Algebras and FL$$^2$$-algebras

Galatos

Jipsen

2020

Lecture Notes in Computer Science

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FL -algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL -algebras. We prove that the congruences of FL -algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras. For a positive idempotent element p in a relation algebra , the double division conucleus image is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class of subalgebras of is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies , and we prove that the identity holds only in trivial members of .

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