We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We give the characterization of all unilinear residuated lattices. By presenting the constructions and axiomatizations for different classes of unilinear residuated lattices, we conclude that the study of unilinear residuated lattices can be reduced to the study of the ⊤-unital ones. Using the classification of unilinear residuated lattices, the idempotent unilinear residuated lattices are studied and amalgamation property and strong amalgamation properties are proved or disproved, depending on if there are extra constants in the language. We give two general constructions of ⊤-unital unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite embeddability property for various subvarieties. Finally, we study the involutive unilinear residuated lattices and give the characterization of a class of commutative 1-involutive compact unilinear residuated lattices. We present some open problems and future work at the end.
","abstract_html":"<p>We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We give the characterization of all unilinear residuated lattices. By presenting the constructions and axiomatizations for different classes of unilinear residuated lattices, we conclude that the study of unilinear residuated lattices can be reduced to the study of the ⊤-unital ones. Using the classification of unilinear residuated lattices, the idempotent unilinear residuated lattices are studied and amalgamation property and strong amalgamation properties are proved or disproved, depending on if there are extra constants in the language. We give two general constructions of ⊤-unital unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite embeddability property for various subvarieties. Finally, we study the involutive unilinear residuated lattices and give the characterization of a class of commutative 1-involutive compact unilinear residuated lattices. We present some open problems and future work at the end.</p>","abstract_has_math":false,"creators":["Zhuang, Xiao"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Nikolaos Galatos","Michael Kinyon","Andrew Linshaw","Mark Siemens","Petr Vojtechovsky"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-08-01T07:00:00Z","date_published":"2023-08-01T07:00:00Z","updated_at":"2026-07-24T02:01:39Z","subjects":["Amalgamation","Axiomatization","Classification","Involutive","Subvarieties","Unilinear residuated lattices","Algebra","Mathematics","Other Mathematics","Physical Sciences and Mathematics"],"languages":["English (eng)"],"rights":["Copyright is held by the author. User is responsible for all copyright compliance.
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"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Unilinear Residuated Lattices"]}]}],"canonical_facts":{"dc:contributor":["Nikolaos Galatos","Michael Kinyon","Andrew Linshaw","Mark Siemens","Petr Vojtechovsky"],"dc:creator":["Zhuang, Xiao"],"dc:date.available":["2024-09-12T07:00:00Z"],"dc:description.abstract":["We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We give the characterization of all unilinear residuated lattices. By presenting the constructions and axiomatizations for different classes of unilinear residuated lattices, we conclude that the study of unilinear residuated lattices can be reduced to the study of the ⊤-unital ones. Using the classification of unilinear residuated lattices, the idempotent unilinear residuated lattices are studied and amalgamation property and strong amalgamation properties are proved or disproved, depending on if there are extra constants in the language. We give two general constructions of ⊤-unital unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite embeddability property for various subvarieties. Finally, we study the involutive unilinear residuated lattices and give the characterization of a class of commutative 1-involutive compact unilinear residuated lattices. We present some open problems and future work at the end.
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"],"dc:subject":["Amalgamation","Axiomatization","Classification","Involutive","Subvarieties","Unilinear residuated lattices","Algebra","Mathematics","Other Mathematics","Physical Sciences and Mathematics"],"dc:title":["Unilinear Residuated Lattices"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T02:01:39Z"}