Lattice Of Closed Sets Research Articles

REFLEXIVITY INDEX AND IRRATIONAL ROTATIONS

AbstractWe determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.

On two problems about sobriety of topological spaces

In this paper, we answer two problems concerning sobriety. One concerns the existence of a closed set lattice whose Scott space is non-sober. Another is about the reflectivity of a category of k-bounded sober spaces. We also prove some sufficient conditions for the Scott space of a dcpo to be k-bounded well-filtered.

Lattices of Irreducibly-derived Closed Sets

This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set ΓSI(X) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of CSI-continuous posets and CSI-prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y, X≅Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.

Cσ-unique Dcpos and Non-maximality of the Class of Dominated Dcpos Regarding Γ-faithfulness

We continue the study of the dcpos which are determined by their Scott closed set lattices. Such dcpos are called Cσ-unique. Some new sufficient conditions for a dcpo to be Cσ-unique are given. One example is constructed to show that a Cσ-unique dcpo need not be dominated. Thus the question whether the dominated dcpos form a maximal Γ-faithful class of dcpos is negatively answered. Using a recent result by Zhao and Xi, we also deduce that every T1 topological space has a dcpo model that is Cσ-unique.

Lattices, closures systems and implication bases: A survey of structural aspects and algorithms

Concept lattices and closed set lattices are graphs with the lattice property. They have been increasingly used this last decade in various domains of computer science, such as data mining, knowledge representation, databases or information retrieval. A fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table. A consequence is the existence of a bijective link between lattices, contexts (via the table) and a set of implicational rules (via the canonical (direct) basis). The possible transformations between these objects give rise to relevant tools for data analysis.In this paper, we present a survey of lattice theory, from the algebraic definition of a lattice, to that of a concept lattice, through closure systems and implicational rules; including the exploration of fundamental bijective links between lattices, reduced contexts and bases of implicational rules; and concluding with the presentation of the main generation algorithms of these objects.

Closed-set lattice and modular matroid induced by covering-based rough sets

Covering is a common form of data representation, and covering-based rough sets, a technique of granular computing, provide an effective tool to deal with this type of data. However, many important problems of covering-based rough sets, such as covering reduction, are NP-hard so that most algorithms to solve them are greedy ones. Matroid theory, based on linear algebra and graph theory, provides well-established platforms for greedy algorithms. Lattice has been widely used in diverse fields, especially algorithm design, which plays an important role in covering reduction. Therefore, it is necessary to integrate covering-based rough sets with matroid and lattice. In this paper, we construct three types of matroids through covering-based rough sets and investigate their modularity. Moreover, we investigate some characteristics of these types of closed-set lattices induced by these three types of matroids and the relationships among these closed-set lattices. First, based on covering-based rough sets, three families of sets are constructed and proved to satisfy independent set axiom of matroids. So three types of matroids are induced by covering-based rough sets in this way. Second, some characteristics of these matroids, such as rank function, closure operator and closed set, are presented. Moreover, we investigate the characteristics of these closed-set lattices induced by these three types of matroids, such as modular pair, modular element. Finally, the relationships among these closed-set lattices induced by these three types of matroids are investigated. Especially, we prove that these three types of matroids induced by covering-based rough sets are all modular matroids.

Closed-set lattice of regular sets based on a serial and transitive relation through matroids

Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a serial and transitive relation on a universe, the collection of all the regular sets of the generalized rough set is a lattice. In this paper, we use the lattice to construct a matroid and then study relationships between the lattice and the closed-set lattice of the matroid. First, the collection of all the regular sets based on a serial and transitive relation is proved to be a semimodular lattice. Then, a matroid is constructed through the height function of the semimodular lattice. Finally, we propose an approach to obtain all the closed sets of the matroid from the semimodular lattice. Borrowing from matroids, results show that lattice theory provides an interesting view to investigate rough sets.

Semi-supervised learning on closed set lattices

We propose a new approach for semi-supervised learning using closed set lattices, which have been recently used for frequent pattern mining within the framework of the data analysis technique of Formal Concept Analysis FCA. We present a learning algorithm, called SELF SEmi-supervised Learning via FCA, which performs as a multiclass classifier and a label ranker for mixed-type data containing both discrete and continuous variables, while only few learning algorithms such as the decision tree-based classifier can directly handle mixed-type data. From both labeled and unlabeled data, SELF constructs a closed set lattice, which is a partially ordered set of data clusters with respect to subset inclusion, via FCA together with discretizing continuous variables, followed by learning classification rules through finding maximal clusters on the lattice. Moreover, it can weight each classification rule using the lattice, which gives a partial order of preference over class labels. We illustrate experimentally the competitive performance of SELF in classification and ranking compared to other learning algorithms using UCI datasets.

On the first-order expressibility of lattice properties related to unicoherence in continua

Many properties of compacta have textbook definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to the existence of holes in a continuum and in certain of its subcontinua.

Dcpo-completion of posets

We introduce a new type of dcpo-completion of posets, called D -completion. For any poset P , the D -completion exists, and P and its D -completion have the isomorphic Scott closed set lattices. This completion is idempotent. A poset P is continuous (algebraic) if and only if its D -completion is continuous(algebraic). Using the D -completion, we construct the local dcpo-completion of posets, that revises the one given by Mislove. In the last section, we define and study bounded sober spaces.

Analogizing Hutton's quasi-uniformities for complete lattices and extending Shi's quasi-uniformities to closed set lattices

This paper gives analogies of Hutton's quasi-uniformities for complete lattices and extensions of Shi's pointwise quasi-uniformities to closed set lattices. CL GOH denotes the category of all complete lattices with generalized order homomorphisms as morphisms and CSL the category of all closed set lattices. HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices and Shi pointwise quasi-uniform spaces on closed set lattices, respectively. We prove that both HQUnif and SPQUnif are fibre-complete topological categories over CL GOH and CSL with respect to the expected forgetful functors, respectively. Also, a mistake in W. Kotzé [Uniform spaces, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 553–580] is pointed out.

Some Properties of Categories CL and CLPair

Structures of products, coproducts, subobjects, extremal subobjects, quotient objects and extremal quotient objects in CL (the category of closed-set lattices) and CLPair (the category of closed-set lattice pairs) are given.

Efficient mining of association rules using closed itemset lattices

Discovering association rules is one of the most important task in data mining. Many efficient algorithms have been proposed in the literature. The most noticeable are Apriori, Mannila's algorithm, Partition, Sampling and DIC, that are all based on the Apriori mining method: pruning the subset lattice (itemset lattice). In this paper we propose an efficient algorithm, called Close, based on a new mining method: pruning the closed set lattice (closed itemset lattice). This lattice, which is a sub-order of the subset lattice, is closely related to Wille's concept lattice in formal concept analysis. Experiments comparing Close to an optimized version of Apriori showed that Close is very efficient for mining dense and/or correlated data such as census style data, and performs reasonably well for market basket style data.

On the lower length of the closed-set lattice of a tree

Let L( T) be the closed-set latice of a tree T. The lower length l ∗(L(T)) of L( T) is defined as l ∗(L(T))= min≈{;|Γ|-1: Γ is a maximal chain in L(T) ≈}; Call a set S of vertices in T a sparse set if d( x, y) ⩾ 3 for any two distinct vertices x, y in S. The sparsity γ( T) of T is defined as γ( T) = max ≈{;| S|: S is a sparse set of T≈}; We prove that, for any tree T of order n, l ∗(L(T)) = n + 1 − γ(T) and deduce from this that l ∗(L(T)) ⩾[n/2] + 1 . All trees T of order n such that l ∗(L(T)) = [n/2] + 1 are characterized.

On the Sequence of Closed‐Set Lattices of a Graph

On the Sequence of Closed‐Set Lattices of a Graph

Constructions of sensitive graphs which are not strongly sensitive

Publisher Summary This chapter describes constructions of sensitive graphs, which are not strongly sensitive. It introduces various classes of graphs in terms of their closed-set lattices. While strongly sensitive graphs are abundant, not every graph is maximally critical or minimally critical. There exist maximally critical graphs that are not minimally critical and vice versa. The chapter introduces some methods of construction that produce various families of sensitive graphs that are not strongly sensitive. All graphs are assumed to be finite, nontrivial, and simple. The chapter also describes the construction of a class of sensitive rambutans that are not strongly sensitive.

Products of graphs with their closed-set lattices

Let L ( G), V( G) and Ḡ be, respectively, the closed-set lattice, vertex set, edge set and complement of a graph G. Any lattice isomorphism Φ: L(G)⋍ L(G′) induces a bijection Φ: V( G)→ V( G′) such that for each x in V( G), Φ( x)= x′ iff Φ({ x})={ x′}. A graph G is strongly sensitive if for any graph G′ and any lattice isomorphism Φ: L(G)⋍ L(G′) , the bijection Φ induced by Φ is a graph isomorphism of G onto G′. G is minimally critical if L(G) ∥ L(G-e) for each e in E( G), and maximally critical if L(G) ∥ L(G+e) for any e in E( G ̆ ) . In this paper, we prove that for any two nontrivial graphs G 1 and G 2, (1) G 1 × G 2 is maximally critical, and (2) G 1 × G 2 is strongly sensitive iff G 1 × G 2 is minimally critical. Necessary and sufficient conditions on G 1 such that G 1 × G 2 is strongly sensitive are also obtained.

On the uniformity of the closed-set lattice of a tree

Denote by l ∗(L) and l ∗(L) respectively the upper length and lower length of a finite lattice L. The lattice L is said to be uniform if for each integer k with l ∗(L)<k< l ∗(L) there exists in L a maximal chain of length k. It is shown that the closed-set lattice of a finite graph G is uniform if G is a tree. The result is not necessarily true if G is not a tree.

On an isomorphism problem on the closed-set lattice of a graph

Let ℒ(G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism Φ: ℒV(G)≃ℒ(G′) induces a bijection ϕ: V(G)→V(G′) such that for each x in V(G), ϕ(x)=x' in V(G') iff Φ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism Φ: ℒ(G)≃ℒ(G′), the bijection ϕ induced by Φ is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C4-free graphs and all covering graphs of finite lattices are strongly sensitive.