Document Type
Conference Proceeding
Publication Date
8-12-2024
Abstract
A residuated poset is a structure ⟨A,⩽, ·, \, /, 1⟩ where ⟨A,⩽⟩ is a poset and ⟨A, ·, 1⟩ is a monoid such that the residuation law x · y ⩽ z ⇐⇒ x ⩽ z/y ⇐⇒ y ⩽ x\z holds. A residuated poset is balanced if it satisfies the identity x\x ≈ x/x. By generalizing the well-known construction of Płonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two families of maps (instead of one, as in the usual case), we construct a residuated poset based on the disjoint union of their domains. We apply this approach to provide a structural description of some varieties of residuated lattices and relation algebras.
Recommended Citation
Bonzio, S., Gil-Férez, J., Jipsen, P., Přenosil, A., Sugimoto, M. (2024). On the Structure of Balanced Residuated Partially Ordered Monoids. In: Fahrenberg, U., Fussner, W., Glück, R. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2024. Lecture Notes in Computer Science, vol 14787. Springer, Cham. https://doi.org/10.1007/978-3-031-68279-7_6
Copyright
The authors
Comments
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Relational and Algebraic Methods in Computer Science. RAMiCS 2024, Lecture Notes in Computer Science, volume 14787, in 2024 following peer review. The final publication may differ and is available at Springer via https://doi.org/10.1007/978-3-031-68279-7_6.