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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.