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Congruence Lattices of Finite Algebras

Congruence Lattices of Finite Algebras

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2012

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DeMeo, William J.

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Freese, Ralph

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University of Hawaii at Manoa

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##### Abstract

An important and long-standing open problem in universal algebra asks whether every finite lattice
is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our
understanding of finite algebras is incomplete, since, given an arbitrary finite algebra, we cannot say
whether there are any restrictions on the shape of its congruence lattice. If we find a finite lattice
that does not occur as the congruence lattice of a finite algebra (as many suspect we will), then we
can finally declare that such restrictions do exist.
By a well known result of Palfy and Pudlak, the problem would be solved if we could prove
the existence of a finite lattice that is not the congruence lattice of a transitive group action or,
equivalently, is not an interval in the lattice of subgroups of a finite group. Thus the problem of
characterizing congruence lattices of finite algebras is closely related to the problem of characterizing
intervals in subgroup lattices.
In this work, we review a number of methods for finding a finite algebra with a given congruence
lattice, including searching for intervals in subgroup lattices. We also consider methods for proving
that algebras with a given congruence lattice exist without actually constructing them. By combining
these well known methods with a new method we have developed, and with much help from computer
software like the UACalc and GAP, we prove that with one possible exception every lattice with at
most seven elements is isomorphic to the congruence lattice of a finite algebra. As such, we have
identified the unique smallest lattice for which there is no known representation. We examine this
exceptional lattice in detail, and prove results that characterize the class of algebras that could
possibly represent this lattice.
We conclude with what we feel are the most interesting open questions surrounding this problem
and discuss possibilities for future work.

##### Description

Thesis (Ph. D.)--University of Hawaii at Manoa, 2012.

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viii, 122 leaves

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##### Related To

Theses for the degree of Doctor of Philosophy (University of Hawaii at Manoa). Mathematics ; no. ????

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