Finite semilattices with many congruences

For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.Comment: 14 pages, 4 figure

Lattices with many congruences are planar

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.Comment: 10 pages, 2 figure

On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them

We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if $n\geq 7$ is a natural number, then the four largest numbers of congruences of the $n$--element (dual) weakly complemented lattices are: $2^{n-2}+1$, $2^{n-3}+1$, $5\cdot 2^{n-6}+1$ and $2^{n-4}+1$. For smaller numbers of elements, several intermediate numbers of congruences appear between the elements of this sequence. After determining these numbers, along with the structures of the (dual) weakly complemented lattices having these numbers of congruences, we derive a similar result for weakly dicomplemented lattices.Comment: 28 page

Planar semilattices and nearlattices with eighty-three subnearlattices

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp.Comment: 71 pages, 7 figure

Lattices of congruence relations for inverse semigroups

The study of congruence relations is acknowledged as fundamental to the study of algebras. Inverse semigroups are a widely studied class for which congruences are well understood. We study one sided congruences on inverse semigroups. We develop the notion of an inverse kernel and show that a left congruence is determined by its trace and inverse kernel. Our strategy identifies the lattice of left congruences as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. This is a natural way to describe one sided congruences with many desirable properties, including that a pair is the inverse kernel and trace of a left congruence precisely when it is the inverse kernel and trace of a right congruence. We classify inverse semigroups for which every Rees left congruence is finitely generated, and provide alternative proofs to classical results, including classifications of left Noetherian inverse semigroups, and Clifford semigroups for which the lattice of left congruences is modular or distributive. In the second half of this thesis we study the partial automorphism monoid of a finite rank free group action. Congruences are described in terms a Rees congruence, subgroups of direct powers of the group and a subgroup of the wreath product of the group and a symmetric group. Via analysis of the subgroups arising in this description we show that, for finite groups, the number of congruences grows polynomially in the rank of action with an exponent related to the chief length of the group