On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity. In previous work, it has been proven that, for any mapping f : L → [special characters omitted] between two complete lattices L and [special characters omitted], there exists a largest residuated mapping ρf dominated by f, and the notion of the shadow σ f of f is introduced. A complete lattice [special characters omitted] is completely distributive if, and only if, the shadow of any mapping f : L → [special characters omitted] from any complete lattice L to [special characters omitted] is residuated. Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice. We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → [special characters omitted] between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = {[special characters omitted][x] | x ∈ L} and fo = [special characters omitted]. The umbral number for any isotone mapping f is equal to the umbral number for fo, and [special characters omitted] for any ordinal number α. Let [special characters omitted] be the maximal umbral number for all isotone mappings f : L → [special characters omitted] between two complete lattices. If L is a ∼ finite lattice, then [special characters omitted]. The computation of [special characters omitted] is less than or equal to that of [special characters omitted], we have a more efficient method to calculate the umbral number [special characters omitted]. The previous results indicate that the umbral number [special characters omitted] determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width

Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.Comment: This version: updated figures and added discussion of generalised time quasilattices. 17 pages, 4 figure

Results in lattices, ortholattices, and graphs

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce π-versions of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order- skeleton of a lattice L to be L˜ := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D0π, ( ii) L˜ satisfies any of the five π-versions of distributivity, (iii) the order-skeleton L˜ is distributive. Second, the symmetric difference notion for Boolean algebra is well-known. Matoušek introduced the orthocomplemented difference lattices (ODLs), which are ortholattices associated with a symmetric difference. He proved that the class of ODLs forms a variety. We focus on the class of all ODLs that are set-representable and prove that this class is not locally finite by constructing an infinite set-representable ODL that is generated by three elements. In the graph theory part, we prove generating theorems and splitter theorems for 5-regular graphs. A generating theorem for a certain class of graphs tells us how to generate all graphs in this class from a few graphs by using some graph operations. A splitter theorem tells us how to build up any graph G from any graph H if G contains H. In this dissertation, we find generating theorems for 5-regular graphs and 5-regular loopless graphs for various edge-connectivities. We also find splitter theorems for 5-regular graphs for various edge-connectivities

The critical end point of QCD

We investigate the critical end point of QCD with two flavours of light dynamical quarks at finite lattice cutoff a=1/4T using a Taylor expansion of the baryon number susceptibility. We find a strong volume dependence of the position of the critical end point. In the large volume limit we obtain T^E/T_c \~ 0.95 and mu_B^E/T^E ~ 1.1, where T_c is the cross over temperature at zero chemical potential, and T^E and mu_B^E are the temperature and the baryon chemical potential at the critical end point. The small value of mu_B^E places it in the range of observability in energy scans at the RHIC.Comment: this version contains all the figures that appear in the published journal versio