Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

Characterizations of Complete Sublattices of a Given Complete Lattice

To characterize all complete sublattices of a given complete lattice is an important problem. In this paper we will give three different characterizations of complete sublattices of a complete lattice by using closure operators, kernel operators, and by using Galoisclosed subrelations.The research was supported by a grant of the faculty of Science ChiangMai University Thailand.     

n-closure systems and n-closure operators

It is very well known and permeating the whole of mathematics that a closure operator on a given set gives rise to a closure system, whose constituent sets form a complete lattice under inclusion, and vice-versa. Recent work of Wille on triadic concept analysis and subsequent work by the author on polyadic concept analysis led to the introduction of complete trilattices and complete n-lattices, respectively, that generalize complete lattices and capture the order-theoretic structure of the collection of concepts associated with polyadic formal contexts. In the present paper, polyadic closure operators and polyadic closure systems are introduced and they are shown to be in a relationship similar to the one that exists between ordinary (dyadic) closure operators and ordinary (dyadic) closure systems. Finally, the algebraic case is given some special consideration. This paper is dedicated to Walter Taylor. Received March 10, 2005; accepted in final form March 7, 2006.     

Weakening Additivity in Adjoining Closures

In this paper, we weaken the conditions for the existence of adjoint closure operators, going beyond the standard requirement of additivity/co-additivity. We consider the notion of join-uniform (lower) closure operators, introduced in computer science, in order to model perfect lossless compression in transformations acting on complete lattices. Starting from Janowitz’s characterization of residuated closure operators, we show that join-uniformity perfectly weakens additivity in the construction of adjoint closures, and this is indeed the weakest property for this to hold. We conclude by characterizing the set of all join-uniform lower closure operators as fix-points of a function defined on the set of all lower closures of a complete lattice.     

Joins in the Frame of Nuclei

Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixed-point theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the technique, we offer a simple (and also intuitionistic) proof of the localic Hofmann–Mislove Theorem.     

Anamorphoses and Flat Morphological Operators on Power Lattices

Flat morphological operators are operators on grey-level images derived from increasing set operators by a combination of thresholding and stacking. For analog grey-levels, they commute with anamorphoses or contrast mappings, that is, continuous increasing grey-level transformations; when the underlying set operator is upper semi-continuous, they also commute with thresholding. For bounded discrete grey-levels, commutation with increasing grey-level transformations and with thresholding is guaranteed, without any continuity conditions. In this paper we consider flat operators for images defined on an arbitrary space of points and taking their values in an arbitrary complete lattice. We study their commutation with increasing transformations of values. This requires some continuity requirements on the transformations of values or on the underlying set operator, which are expressed in terms of the lattice of values. We obtain as particular cases the known conditions for analog and discrete grey-levels, and also new conditions for other examples of values: multivalued vectors or any finite set of values.     

Algebraic axiomatization of tense intuitionistic logic

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.     

Upper semicontinuity of set-valued functions

Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.     

Fuzzy logic, continuity and effectiveness

 It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general). Received: 15 February 2001 / Revised version: 31 May 2001 / Published online: 12 July 2002     

Closure Operators and Lattice Extensions

For closure operators Γ and Δ on the same set X, we say that Δ is a weak (resp. strong) extension of Γ if Cl(X, Γ) is a complete meet-subsemilattice (resp. complete sublattice) of Cl(X, Δ). This context is used to describe the extensions of a finite lattice that preserve various properties. This revised version was published online in September 2006 with corrections to the Cover Date.     

Measures on spaces of operators and isometries

One considers a question that arises at the investigation of isometric operators in vector-valued L^p -spaces. Let E, F be Banach spaces, let p>0, let μ be a probability Borel measure on the space on continuous linear operators from E into F such that for any e ε E one has $$\left\| e \right\|^p = \int {\left\| {Te} \right\|^p } d\mu \left( T \right)$$ . In the cases when: 1) E=C(K), K is a metric compactum, F is an arbitrary space, p>1 and 2) 2)E=F=L^q,p>1, q>1 q ?[p,2] it is Proved that the support of the measure μ is contained in the set of the operators that are scalar multiples of isometries. For E=C(K) one obtains an isomorphic analogue of this result: if the Banach-Mazur distance between C(K) and the P -sum of Banach spaces is small, then the distance between C(K) and one of the spaces is small.     

Dynamic effect algebras

We introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.     

An order-theoretic perspective on categorial closure operators

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.     

完全分配格上的覆盖粗糙集模型

将集合论中的覆盖概念抽象到完全分配格L上,利用它定义格L上关于覆盖的上(下)近似算子,给出格L上覆盖粗糙集模型.文中先讨论格L上覆盖的相关性质,进而研究了覆盖上(下)近似算子的性质,得到若干结果.     

Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices

Order - Given a k-tuple P=(x 1,x 2,...,x k ) in a finite lattice X endowed with the lattice metric d, a median of P is an element m of X minimizing the sum ∑ i d(m,x i ). If X is an upper...     

Weak relative pseudo-complements of closure operators

We define the notion of weak relative pseudo-complement on meet semi-lattices, and we show that it is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main result is that if a complete lattice is meet-continuous, then every closure operator on admits weak relative pseudo-complements with respect to continuous closure operators on .Presented by E. T. Schmidt.     

Bands in lattices of operators

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.

     

On Finite Elements in Lattices of Regular Operators

Let E and F be vector lattices and the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in , and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in . We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient conditions for rank one operators to be finite elements in the vector lattice . A half year stay at the Technical University of Dresden was supported by China Scholarship Council.     

Idempotent Block Splitting on Partial Partitions,I: Isotone Operators

Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.     

<Emphasis Type="BoldItalic">κ</Emphasis>-Complete Uniquely Complemented Lattices

We show that for any infinite cardinal κ, every complete lattice where each element has at most one complement can be regularly embedded into a uniquely complemented κ-complete lattice. This regular embedding preserves all joins and meets, in particular it preserves the bounds of the original lattice. As a corollary, we obtain that every lattice where each element has at most one complement can be embedded into a uniquely complemented κ-complete lattice via an embedding that preserves the bounds of the original lattice.