Gerhard Gierz

A Compendium of continuous lattices

O. A Primer of Complete Lattices.- 1. Generalities and notation.- 2. Complete lattices.- 3. Galois connections.- 4. Meet-continuous lattices.- I. Lattice Theory of Continuous Lattices.- 1. The way-below relation.- 2. The equational characterization.- 3. Irreducible elements.- 4. Algebraic lattices.- II. Topology of Continuous Lattices: The Scott Topology.- 1. The Scott topology.- 2. Scott-continuous functions.- 3. Injective spaces.- 4. Function spaces.- III. Topology of Continuous Lattices: The Lawson Topology.- 1. The Lawson topology.- 2. Meet-continuous lattices revisited.- 3. Lim-inf convergence.- 4. Bases and weights.- IV. Morphisms and Functors.- 1. Duality theory.- 2. Morphisms into chains.- 3. Projective limits and functors which preserve them.- 4. Fixed point construction for functors.- V. Spectral Theory of Continuous Lattices.- 1. The Lemma.- 2. Order generation and topological generation.- 3. Weak irreducibles and weakly prime elements.- 4. Sober spaces and complete lattices.- 5. Duality for continuous Heyting algebras.- VI. Compact Posets and Semilattices.- 1. Pospaces and topological semilattices.- 2. Compact topological semilattices.- 3. The fundamental theorem of compact semilattices.- 4. Some important examples.- 5. Chains in compact pospaces and semilattices.- VII. Topological Algebra and Lattice Theory: Applications.- 1. One-sided topological semilattices.- 2. Topological lattices.- 3. Compact pospaces and continuous Heyting algebras.- 4. Lattices with continuous Scott topology.- Listof Symbols.- List of Categories.

Dynein and dynactin deficiencies affect the formation and function of the Spitzenkörper and distort hyphal morphogenesis of Neurospora crassa

The impact of mutations affecting microtubule-associated motor proteins on the morphology and cytology of hyphae of Neurospora crassa was studied. Two ropy mutants, ro-1 and ro-3, deficient in dynein and dynactin, respectively, were examined by video-enhanced phase-contrast microscopy and image analysis. In contrast to the regular, hyphoid morphology of wild-type hyphae, the hyphae of the ropy mutants exhibited a great variety of distorted, non-hyphoid morphologies. The ropy hyphae were slow-growing and manifested frequent loss of growth directionality. Cytoplasmic appearance, including organelle distribution and movement, were ostensibly different in the ropy hyphae. The Spitzenkörper (Spk) of wild-type hyphae was readily seen by phase-contrast optics; the Spk of both ro-1 and ro-3 was less prominent and sometimes undetectable. Only the fast-growing ropy hyphae displayed a Spk, and it was smaller and less phase-dark than the wild-type Spk. Growth rate in both wild-type and ropy mutants was directly correlated with the size of the Spk. Spk efficiency, measured in terms of cell area generated per Spk travelled distance, was lower in ropy mutants. Another salient difference between ropy mutants and wild-type hyphae was in Spk trajectory. Whereas the Spk of wild-type hyphae maintained a trajectory close to the cell growth axis, the Spk of ropy hyphae moved much more erratically. Sustained departures in the trajectory of the ropy Spk produced corresponding distortions in hyphal morphology. A causal correlation between Spk trajectory and cell shape was tested with the Fungus Simulator program. The characteristic morphologies of wild-type or ropy hyphae were reproduced by the Fungus Simulator, whose vesicle supply centre (VSC) was programmed to follow the corresponding Spk trajectories. This is evidence that the Spk controls hyphal morphology by operating as a VSC. These findings on dynein or dynactin deficiency support the notion that the microtubular cytoskeleton plays a major role in the formation and positioning of the Spk, with dramatic consequences on hyphal growth and morphogenesis.

Markovian statistics on evolving systems

A novel framework for the analysis of observation statistics on time discrete linear evolutions in Banach space is presented. The model differs from traditional models for stochastic processes and, in particular, clearly distinguishes between the deterministic evolution of a system and the stochastic nature of observations on the evolving system. General Markov chains are defined in this context and it is shown how typical traditional models of classical or quantum random walks and Markov processes fit into the framework and how a theory of quantum statistics (sensu Barndorff-Nielsen, Gill and Jupp) may be developed from it. The framework permits a general theory of joint observability of two or more observation variables which may be viewed as an extension of the Heisenberg uncertainty principle and, in particular, offers a novel mathematical perspective on the violation of Bell’s inequalities in quantum models. Main results include a general sampling theorem relative to Riesz evolution operators in the spirit of von Neumann’s mean ergodic theorem for normal operators in Hilbert space.

Level sets in finite distributive lattices of breadth 3

Abstract A level set in a distributive lattice consists of all elements of a certain fixed rank r. In this note we give a characterization of maximal level sets in distributive lattices of breadth 3. Several applications are given, one of which shows that if a distributive lattice L of breadth 3 is generated by a level set A of maximal cardinality, then A consists of all elements for which the rank is equal 1 2 times the rank of the largest element of L.

ESSENTIAL COMPLETIONS OF DISTRIBUTIVE LATTICES

The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the same way as the one-point compactification of a locally compact topological space does in it s setting.

Spectral Theory of Continuous Lattices

Opectral theory plays an important and well-known role in such areas as the theory of commutative rings, lattices, and of C*-algebras, for example. The general idea is to define a notion of “prime element” (more often: ideal element) and then to endow the set of these primes with a topology. This topological space is called the “spectrum” of the structure. One then seeks to find how algebraic properties of the original structure are reflected in the topological properties of the spectrum; in addition, it is often possible to obtain a representation of the given structure in a concrete and natural fashion from the spectrum.

Morphisms and Functors

With the exception of certain developments in Chapter II, notably Sections 2 and 4, we largely refrained from using category-theoretic language (even when we used its tools in the context of Galois connections). Inevitably, we have to consider various types of functions between continuous lattices, and this is a natural point in our study to use the framework of category theory.

A Primer on Complete Lattices

This introductory chapter serves as a convenient source of reference for certain basic aspects of complete lattices needed in the sequel. The experienced reader may wish to skip directly to Chapter I and the beginning of the discussion of the main topic of this book: continuous lattices, a special class of complete lattices.

Topology of Continuous Lattices: The Lawson Topology

The first topologies defined on a lattice directly from the lattice ordering (that is, Birkhoffs order topology and Frink’s interval topology) involved “symmetrical” definitions—the topologies assigned to L and to Lop were identical. The guiding example was always the unit interval of real numbers in its natural order, which is of course a highly symmetrical lattice. The initial interest was in such questions as which lattices became compact and/or Hausdorff in these topologies. The Scott topology stands in strong contrast to such an approach. Indeed it is a “one-way” topology, since, for example, all the open sets are always upper sets; thus, for nontrivial lattices, the T0-separation axiom is the strongest it satisfies. Nevertheless, we saw in Chapter II that the Scott topology provides many links between continuous lattices and general topology in such classical areas as the theory of semicontinuous functions and in the study of lattices of closed (compact, convex) sets (ideals) in many familiar structures.

Topological Algebra and Lattice Theory: Applications

Our last chapter is devoted to exploring further links between topological algebra and continuous lattices. This theme has already played an important role: the Fundamental Theorem of Compact Semilattices (VI-3.4) is just one example. In this chapter, however, the methods of topological algebra occupy a more central role, while the methods of continuous lattices are somewhat less prominent.