# In memoriam William Lampe

Emeritus faculty member William (Bill) Austyn Lampe died on January 8th 2023, surrounded by family.

Except for a one year post doc at Colorado and a two year stay at the Institute for Advanced Study, he spent his whole career at UH. He came here in 1972, was chair 1983-1986, and retired in 2011.
He will be missed.

#### Mathematical research

A major theme of Bill’s work was congruence lattice representations, that is, the problem of finding an algebra whose family of congruence relations is isomorphic to a given lattice. For example, the three-element lattice $\mathbf{0}<\mathbf{1}<\mathbf{2}$ arises as the congruence lattice of the symmetric group $S_3$.

Grätzer and Schmidt in 1963 bridged lattice theory and universal algebra by showing that every compactly generated lattice has such a representation. Simplifications were found by Lampe (1973) and Pudlak (1976). They involved what was affectionally known as the sack-of-potatoes construction, pictured above. Such simplifications turned out to lead to applications in other mathematical fields. Bill used his methods to develop what have become known as the Zipper Lemma and the Term Condition.

As well as these positive results, he had several striking negative results. For example,
a longstanding conjecture was that every algebraic lattice could be represented as the congruence lattice of an algebra having only one operation. Bill showed that there are lattices such that the algebra must have at least $\kappa$ many operations, for any cardinal $\kappa$.

In Bill’s last paper, The strength of the Grätzer-Schmidt theorem (with Brodhead, Khan, Kjos-Hanssen, Nguyen and Shore, 2016) the GS theorem is studied in the context of reverse mathematics. As a follow-up, Jack Yoon showed in his 2020 dissertation that surprisingly the Grätzer-Schmidt theorem is provable in the system Arithmetic Transfinite Recursion of reverse mathematics.

After working on many other problems in universal algebra and lattice theory through his career, Bill returned to this topic for his talk at the conference Algebra and Lattices in Hawaii in 2018, held in honor of him and his colleagues in lattice theory and universal algebra in Hawaii.

# Kazuhisa Nakasho’s slides

nakasho-2021-11-17 Slides for Kazuhisa Nakasho’s logic seminar talk.

The Department of Mathematics at University of Hawaii at Manoa has long had an informal graduate program in logic, lattice theory, and universal algebra (People, Courses, Description) going back to Alfred Tarski’s 1963 student William Hanf.

During Fall 2022 the logic seminar had talks by three outside speakers from Chiba, Yamaguchi and Wisconsin.

We are offering the following course rotation (courses mostly repeating after two years):

Past offerings
Semester Course number Course title Instructor
Spring 2016 MATH 649 Applied Model Theory Ross
Fall 2016 MATH 654 Graduate Introduction to Logic Beros
Spring 2017 MATH 657 Computability and Complexity Khan
Spring 2018 MATH 649 Applied Model Theory Ross
Fall 2018 MATH 654 Graduate Introduction to Logic Kjos-Hanssen
Spring 2019 MATH 655 Set theory Williams
Spring 2020 MATH 657 Computability and Complexity Kjos-Hanssen
Fall 2020 MATH 654 Graduate Introduction to Logic Kjos-Hanssen
Spring 2021 MATH 649 Applied model theory Ross
Spring 2022 MATH 657 Computability and Complexity Kjos-Hanssen
Fall 2022 MATH 654 Graduate Introduction to Logic Kjos-Hanssen

Future offerings:

Semester Course number Course title Instructor
Spring 2023 MATH 649B Applied model theory Ross
Spring 2024 MATH 657 Computability and Complexity Kjos-Hanssen

It is also recommended that students familiarize themselves with undergraduate level logic, which is offered on the following schedule:

Past offerings
Semester Course number Course title Instructor
Fall 2012 MATH 454 Axiomatic Set Theory Kjos-Hanssen
Spring 2013 MATH 455 Mathematical Logic Kjos-Hanssen
Fall 2014 MATH 454 Axiomatic Set Theory Ross
Spring 2015 MATH 455 Mathematical Logic Khan
Spring 2016 MATH 454 Axiomatic Set Theory Khan
Spring 2017 MATH 455 Mathematical Logic Ross
Spring 2018 MATH 455 Mathematical Logic Khan
Fall 2019 MATH 454 Axiomatic Set Theory Williams
Spring 2020 MATH 455 Mathematical Logic Williams
Fall 2021 MATH 454 Axiomatic Set Theory Kjos-Hanssen
Spring 2022 MATH 455 Mathematical Logic Ross

Future offerings:

Semester Course number Course title Instructor
Fall 2023 MATH 455 Mathematical Logic Kjos-Hanssen

#### Faculty teaching in the program

David A. Ross, Professor
Bjørn Kjos-Hanssen, Professor

# New book on Subquasivariety Lattices

Professor Emeritus J.B. Nation, frequent visitors Kira Adaricheva and Jennifer Hyndman, and UH’s Joy Nishida have published a new book:

A Primer of Subquasivariety Lattices uniquely develops universal algebra in languages that may not contain equality.

It presents new results in representations of various types of lattices by subquasivarieties, and illustrates theory through concrete examples.

Part of the book series: Canadian Mathematical Society/CAIMS Books in Mathematics (CMS/CAIMS BM, volume 3)