Related papers: Generalizing Galvin and J\'onsson's Classification…
In 1959, F.Galvin and B.Jonsson characterized distributive sublattices of free lattices in their paper. In this paper, I will create new proofs to a portion of Galvin and J\'onsson's results. Based on these new proofs, I will explore…
This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable…
This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable…
This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable…
A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved.…
In general it is a difficult problem to construct the lattice of submodules $L(M)$ of a given module $M$. In \cite{St} R. P. Stanley outlined a method for constucting a distributive lattice from a knowledge of its join irreducibles. However…
A rotational lattice is a structure (L;\vee,\wedge, g) where L=(L;\vee,\wedge) is a lattice and g is a lattice automorphism of finite order. We describe the subdirectly irreducible distributive rotational lattices. Using J\'onsson's lemma,…
The problem of determining which infinite lattices are (isomorphic to) sublattices of free lattices is in general unsolved and extremely difficult. In this note, we reduce the problem by proving that all locally finite sublattices of free…
We prove that for any free lattice F with at least $\aleph\_2$ generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F. This solves a…
Turing Award winner Juris Hartmanis introduced in 1959 lattices of subspaces of generalized partitions ("partitions of type n"; "geometries" if $n = 2$). Hartmanis states it is "an unsolved problem whether there are any incomplete lattice…
The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and…
A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a…
Let S be a distributive {∨, 0}-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let $\phi$: Con K $\to$ S be a {∨, 0}-homomorphism. Then $\phi$ is,…
Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since…
A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error…
This paper studies the differential lattice, defined to be a lattice $L$ equipped with a map $d:L\to L$ that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications…
Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets…
This article is the second part of an essay dedicated to lattices freely generated by posets within a variety. The first part dealt with four easy varieties while this part is concerned with finitely generated varieties. Here we present a…
Let $L$ be a lattice. We call a congruence relation $\gQ$ of $L$ isoform, if any two congruence classes of $\gQ$ are isomorphic (as lattices). Let us call the lattice $L$ isoform, if all congruences of $L$ are isoform. G. Gr\"atzer and…
Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice IdS of S is both algebraic and dually algebraic.…