Date of Award
8-2023
Document Type
Dissertation
Degree Name
Ph.D.
Organizational Unit
College of Natural Science and Mathematics, Mathematics
First Advisor
Nikolaos Galatos
Second Advisor
Michael Kinyon
Third Advisor
Andrew Linshaw
Fourth Advisor
Mark Siemens
Fifth Advisor
Petr Vojtechovsky
Keywords
Amalgamation, Axiomatization, Classification, Involutive, Subvarieties, Unilinear residuated lattices
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.
Copyright Date
8-2023
Copyright Statement / License for Reuse
All Rights Reserved.
Publication Statement
Copyright is held by the author. User is responsible for all copyright compliance.
Rights Holder
Xiao Zhuang
Provenance
Received from ProQuest
File Format
application/pdf
Language
English (eng)
Extent
149 pgs
File Size
779 KB
Recommended Citation
Zhuang, Xiao, "Unilinear Residuated Lattices" (2023). Electronic Theses and Dissertations. 2335.
https://digitalcommons.du.edu/etd/2335
Discipline
Mathematics