Diagrams in Involutive Residuated Lattices

Author

Isis A. Gallardo, University of DenverFollow

Date of Award

Summer 8-24-2024

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

Petr Vojtěchovský

Fourth Advisor

Mandi Schaeffer Fry

Fifth Advisor

Mohammad Mahoor

Copyright Statement / License for Reuse

All Rights Reserved.

Keywords

Decidability, Diagrams, Generation, Involutive, Residuated lattices

Abstract

First, we show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thereby improving the existing Cayley/Holland style embedding theorem. Using this result, we demonstrate that the variety of all dis tributive lattice-ordered pregroups is generated by the functional algebra on the integers. Additionally, we prove that the equational theory of this variety is decidable.

Next, we establish that DLP is equal to the join of its subvarieties LP_n, where 𝑛 ∈ ℤ⁺, consisting of 𝑛-periodic ℓ-pregroups. We also prove that every algebra in LP_n can be embedded into the subalgebra 𝙵_𝑛(Ω) of 𝑛-periodic elements of 𝙵(Ω), for some integral chain Ω. This representation shows that, for every 𝑛, the variety LP_n is generated by the single algebra 𝙵_𝑛(ℚ ^→⨯ℤ), noting that the chain ℚ ^→⨯ℤ is independent of 𝑛.

We further establish that every algebra in LP_n can be embedded into the wreath product of an ℓ-group and 𝙵_𝑛(ℤ), highlighting the significant role of the simple 𝑛-periodic ℓ-pregroup 𝙵_𝑛(ℤ). We also show that the join of the varieties 𝖵(𝙵_𝑛(ℤ)) equals DLP and thus matches the join of the varieties LP_n, even though 𝖵(𝙵_𝑛(ℤ)) ≠ LP_n for any single 𝑛. In this way, DLP has two different well-behaved approximations. Additionally, we demonstrate that, for every 𝑛, the equational theory of 𝙵_𝑛(ℤ) is decidable. Using the wreath product decomposition, we establish that the equational theory of LP_n is also decidable.

Applying similar strategies, we investigate certain varieties of full weakening relations on a chain and prove the decidability of these varieties. We also present the construction of a potential non-distributive ℓ-pregroup and provide preliminary results towards proving this claim.

Copyright Date

8-2024

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Isis A. Gallardo

Provenance

Received from Author

File Format

application/pdf

Language

English (eng)

Extent

155 pgs

File Size

1.1 MB

Recommended Citation

Gallardo, Isis A., "Diagrams in Involutive Residuated Lattices" (2024). Electronic Theses and Dissertations. 2455.
https://digitalcommons.du.edu/etd/2455