Abstract
FL\(^2\)-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\(^2\)-algebras. We prove that the congruences of FL\(^2\)-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 \(\mathbf{A}\), the double division conucleus image \(p{\backslash }\mathbf{A}{/}p\) 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 \(S(\mathsf {dRA})\) of subalgebras of \(\{p{\backslash }\mathbf{A}{/}p:A\in \mathsf {RA}, 1\le p^2=p\in A\}\) is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate \(S(\mathsf {dRA})\) 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 \(0\le 1\), and we prove that the identity \(1\le 0\) holds only in trivial members of \(S(\mathsf {dRA})\).
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Galatos, N., Jipsen, P. (2020). Weakening Relation Algebras and FL\(^2\)-algebras. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_8
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