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Gluing Residuated Lattices

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Abstract

We introduce and characterize various gluing constructions for residuated lattices that intersect on a common subreduct, and which are subalgebras, or appropriate subreducts, of the resulting structure. Starting from the 1-sum construction (also known as ordinal sum for residuated structures), where algebras that intersect only in the top element are glued together, we first consider the gluing on a congruence filter, and then add a lattice ideal as well. We characterize such constructions in terms of (possibly partial) operators acting on (possibly partial) residuated structures. As particular examples of gluing constructions, we obtain the non-commutative version of some rotation constructions, and an interesting variety of semilinear residuated lattices that are 2-potent. This study also serves as a first attempt toward the study of amalgamation of non-commutative residuated lattices, by constructing an amalgam in the special case where the common subalgebra in the V-formation is either a special (congruence) filter or the union of a filter and an ideal.

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Funding

This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 890616 awarded to Ugolini.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, USA

    Nikolaos Galatos

  2. IIIA - CSIC, Bellaterra, Barcelona, Spain

    Sara Ugolini

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  1. Nikolaos Galatos
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  2. Sara Ugolini
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Correspondence to Nikolaos Galatos.

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Galatos, N., Ugolini, S. Gluing Residuated Lattices. Order 40, 623–664 (2023). https://doi.org/10.1007/s11083-023-09626-w

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