Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible elements of $D$, there exists a finite lattice $L$ and an isomorphism from the congruence lattice of $L$ onto $D$ such that $Q$ corresponds to the set of principal congruences of $L$ under this isomorphism. Based on earlier results of G. Gr\"atzer, H. Lakser, and the present author, we prove that a finite distributive lattice $D$ is fully principal congruence representable if and only if it is planar and it has at most one join-reducible coatom. Furthermore, even the automorphism group of $L$ can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Gr\"atzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.Comment: 20 pages, 8 figure

Notes on planar semimodular lattices. I. Construction

We construct all planar semimodular lattices in three simple steps from the direct product of two chains.Comment: 13 pages with 9 diagram

Congruence lattices 101

AbstractThis lecture — based on the author's book, General Lattice Theory, Birkhäuser Verlag, 1978 — briefly introduces the basic concepts of lattice theory, as needed for the lecture “Some combinatorial aspects of congruence lattice representations”

Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem

A new result of G. Cz\'edli states that for an ordered set $P$ with at least two elements and a group $G$, there exists a bounded lattice $L$ such that the ordered set of principal congruences of $L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$. I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1960-s

Varieties of distributive rotational lattices

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, this leads to a description of all varieties of distributive rotational lattices.Comment: 7 page

Reliability of systems with dependent components based on lattice polynomial description

Reliability of a system is considered where the components' random lifetimes may be dependent. The structure of the system is described by an associated "lattice polynomial" function. Based on that descriptor, general framework formulas are developed and used to obtain direct results for the cases where a) the lifetimes are "Bayes-dependent", that is, their interdependence is due to external factors (in particular, where the factor is the "preliminary phase" duration) and b) where the lifetimes' dependence is implied by upper or lower bounds on lifetimes of components in some subsets of the system. (The bounds may be imposed externally based, say, on the connections environment.) Several special cases are investigated in detail