\u3cem\u3en\u3c/em\u3e-Distributivity, Dimension and Carathéodory\u27s Theorem

A. Huhn proved that the dimension of Euclidean spaces can be characterized through algebraic properties of the lattices of convex sets. In fact, the lattice of convex sets of IEn is n+1-distributive but not n-distributive. In this paper his result is generalized for a class of algebraic lattices generated by their completely join-irreducible elements. The lattice theoretic form of Carathédory\u27s theorem characterizes n-distributivity in such lattices. Several consequences of this result are studies. First, it is shown how infinite n-distributivity and Carathédory\u27s theorem are related. Then the main result is applied to prove that for a large class of lattices being n-distributive means being in a variety generated by the finite n-distributive lattices. Finally, n-distributivity is studied for various classes of lattices, with particular attention being paid to convexity lattices of Birkhoff and Bennett for which a Helly type result is also proved

Towards an Algebra for Cascade Effects

We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied models of cascade effects. We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express global systems through combinations of local systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of mu-rank to capture the energy of a system, and understand the minimal amount of effort required to fail a system, termed resilience. We deduce a dual notion of fragility and show that the combination of systems sets a limit on the amount of fragility inherited.Comment: 31 page

The number of meets between two subsets of a lattice

AbstractLet L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove |A| |B| ⩽ |L| |A ∧ B| for any A, B ⊃ L, where |·| denotes cardinality and A ∧ B = {a ∧ b: a ϵ A, b ϵ B}. |A ∧ B| attains its minimum for fixed |A|, |B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed |A|, |B|, |A ∧ B| is minimized when A and B are the |A| and |B| smallest n-digit k-ary numbers written backwards and forwards, respectively. |A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made

Preservation theorems for algebraic and relational models of logic

A thesis submitted to the School of Computer Science, Faculty of Science, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013In this thesis a number of different constructions on ordered algebraic structures are studied. In particular, two types of constructions are considered: completions and finite embeddability property constructions. A main theme of this thesis is to determine, for each construction under consideration, whether or not a class of ordered algebraic structures is closed under the construction. Another main focus of this thesis is, for a particular construction, to give a syntactical description of properties preserved by the construction. A property is said to be preserved by a construction if, whenever an ordered algebraic structure satisfies it, then the structure obtained through the construction also satisfies the property. The first four constructions investigated in this thesis are types of completions. A completion of an ordered algebraic structure consists of a completely lattice ordered algebraic structure and an embedding that embeds the former into the latter. Firstly, different types of filters (dually, ideals) of partially ordered sets are investigated. These are then used to form the filter (dually, ideal) completions of partially ordered sets. The other completions of ordered algebraic structures studied here include the MacNeille completion, the canonical extension (also called the completion with respect to a polarization) and finally a prime filter completion. A class of algebras has the finite embeddability property if every finite partial subalgebra of some algebra in the class can be embedded into some finite algebra in the class. Firstly, two constructions that establish the finite embeddability property for residuated ordered structures are investigated. Both of these constructions are based on completion constructions: the first on the Mac- Neille completion and the second on the canonical extension. Finally, algebraic filtrations on modal algebras are considered and a duality between algebraic and relational versions of filtrations is established

A counterexample to Thiagarajan's conjecture on regular event structures

We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded $\natural$-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects. Namely, from a graph theoretical point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in bijection with CAT(0) cube complexes. A necessary condition for both conjectures to be true is that domains of regular event structures (with bounded $\natural$-cliques) admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded $\natural$-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex whose universal cover is a CAT(0) square complex containing a particular plane with an aperiodic tiling. We prove that other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009). On the positive side, using breakthrough results by Agol (2013) and Haglund and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's conjecture is true for regular event structures whose domains occur as principal filters of hyperbolic CAT(0) cube complexes which are universal covers of finite nonpositively curved cube complexes

Noncommutative lattices

The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began 40 years later with Jonathan Leech\u27s 1989 paper "Skew lattices in rings." Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation, over thirty years ago, of Jonathan Leech\u27s program of research that studied noncommutative variations of lattices. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to the renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as handy reference to researchers of noncommutative algebras