J. B. Nation

A REPORT ON SUBLATTICES OF A FREE LATTICE

Publisher Summary This chapter presents a report on sublattices of a free lattice. It elaborates the characterization problem for finite sublattices of a free lattice. It is conjectured that these are precisely the finite lattices that satisfy three first order conditions, the Whitman condition. Every finite sublattice of a free lattice is an S-lattice, for all three conditions hold in free lattices, and is the key condition in Whitmans famous characterization of free lattices, and is easy consequences of Whitmans results. It is an open question whether, conversely, every S -lattice is embeddable in a free lattice. Several characterizations of finite sublattices of free lattices have been found and some of these even characterize finitely generated sublattices of free lattices, but none of them is in terms of a finite set of first-order formulas. A lattice L is transferable if for every embedding f of L into the ideal lattice of a lattice K , there exists and embedding g of L into K.

Ordered direct implicational basis of a finite closure system

The closure system on a finite set is a unifying concept in logic programming, relational databases and knowledge systems. It can also be presented in the terms of finite lattices, and the tools of economic description of a finite lattice have long existed in lattice theory. We present this approach by defining the D-basis and introducing the concept of an ordered direct basis of an implicational system. A direct basis of a closure operator, or an implicational system, is a set of implications that allows one to compute the closure of an arbitrary set by a single iteration. This property is preserved by the D-basis at the cost of following a prescribed order in which implications will be attended. In particular, using an ordered direct basis allows to optimize the forward chaining procedure in logic programming that uses the Horn fragment of propositional logic. One can extract the D-basis from any direct unit basis @S in time polynomial in the size s(@S), and it takes only linear time of the cardinality of the D-basis to put it into a proper order. We produce examples of closure systems on a 6-element set, for which the canonical basis of Duquenne and Guigues is not ordered direct.

Doubling convex sets in lattices and a generalized semidistributivity condition

A useful construction for lattices is doubling a convex subsetI of a latticeL, i.e., replacingI byI×2. It is shown that this construction preserves a generalized semidistributivity condition (C). Varieties of lattices in which every lattice satisfies (C) are characterized equationally.

On implicational bases of closure systems with unique critical sets

We present results inspired by the study of closure systems with unique critical sets. Many of these results, however, are of a more general nature. Among those is the statement that every optimum basis of a finite closure system, in D. Maiers sense, is also right-side optimum. New parameters for the size of the binary part of a closure system are established. We introduce the K-basis of a closure system, which is a refinement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. The main part of the paper is devoted to closure systems with unique critical sets, and some subclasses of these where the K-basis is unique. A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Consequently, closure systems without D-cycles can be effectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.

Formal descriptions of developing systems

Preface. Acknowledgements. Workshop Photograph. Workshop Participants. Formal Descriptions of Developing Systems J.B. Nation. 1: Global Systems. The Statistical Theory of Global Population Growth S.P. Kapitza. Maximum Reliancy as a Determinant of Food Web Behavior E.A. Laws. Thermodynamic Approach to the Problem of Economic Equilibrium V.M. Sergeev. 2: Biological Systems. Programmed Cell Death Phenomena at Various Levels of Development of the Living Systems V.P. Skulachev. Development of Motor Control in Vertebrates J. Stollberg. Disclocations in the Repetitive Unit Patterns of Biological Systems B. Zagorska-Marek, D. Wiss. The Description of Growth of Plant Organs: A Continuous Approach Based on the Growth Tensor J. Nakielski, Z. Hejnowicz. Using Deterministic Chaos Theory for the Analysis of Sleep EEG J.D. Rand, H.P. Collin, L.E. Kauniai, D.H. Crowell, J. Pearce. 3: Emergence. How Does Complexity Develop? J. Cohen. Adaptive Evolution of Complex Systems under Uncertain Environmental Constraints: A Viability Approach J.-P. Aubin. Archetypal Dynamics: An Approach to the Study of Emergence W.H. Sulis. 4: Modelling. Sociability, Diversity and Compatibility in Developing Systems: EVS Approach I.N. Trofimova. Tetrahymena and Ants- Simple Models of Complex Systems W.A.M. Brandts. Embryogenesis as a Model of a Developing System O.P. Melekhova. 5: Presentations for Discussion. Vagal Tone Chronology in Gulf War Veterans Illnesses and in Acute Stress Reactions H.A. Bracha. Limits of Developing a National System of Agricultural Extension F.H. Arion. An Accumulation Model for the Formation of Mini BlackHoles G.D. Esmer. Subject Index.

The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind–MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. In particular, if K(P) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.

Term Rewrite Systems for Lattice Theory

It is shown that, even though there is a very well-behaved, natural normal form for lattice theory, there is no finite, convergent AC term rewrite system for the equational theory of all lattices.

Discovery of the D-basis in binary tables based on hypergraph dualization

Discovery of (strong) association rules, or implications, is an important task in data management, and it finds application in artificial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a specific set of implications, called the D-basis, that provides a representation for a reduced binary table, based on the structure of its Galois lattice. At the core of the method are the D-relation defined in a lattice theoretic framework and the hypergraph dualization algorithm that allows us to effectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, first developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, artificial intelligence and game theory. One application of the method is for analysis of gene expression data related to a particular phenotypic variable, and some initial testing is done for data provided by the University of Hawaii Cancer Center.

MAXIMAL SUBLATTICES AND FRATTINI SUBLATTICES OF BOUNDED LATTICES

We will address these questions in order, and provide good partial answers, especially for finite lattices which are bounded homomorphic images of a free lattice. Recall that a finite lattice is bounded if and only if it can be obtained from the one element lattice by a sequence of applications of Alan Day’s doubling construction for intervals. In particular, finite distributive lattices are bounded. On the other hand, we do not have a complete solution for any of the above problems. The main results of this paper can be summarized as follows. (1a) For any k > 0, there exists a finite lattice L which has more than |L| maximal sublattices. (1b) A finite bounded lattice L has at most |L| maximal sublattices. (2a) There exist arbitrarily large finite (or even countably infinite) lattices with a maximal sublattice isomorphic to the five element lattice M3. (2b) For any e > 0, there exists a finite bounded lattice L with a maximal sublattice S such that |S| < e|L|. (3a) There exist infinitely many lattice varieties V such that every finite nontrivial lattice L ∈ V is isomorphic to Φ(L′) for some finite lattice L′ ∈ V. (3b) Every finite bounded lattice L can be represented as Φ(K) for some finite bounded lattice K (not necessarily in V(L)).

LATTICES OF PSEUDOVARIETIES

We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all the lattices of subvarieties of varieties V(A) generated by algebras A€. P.