Continuous Lattices Research Articles

Systems of fixpoint equations: Abstraction, games, up-to techniques and local algorithms

Systems of fixpoint equations over complete lattices, consisting of (mixed) least and greatest fixpoint equations, allow one to express many verification tasks such as model-checking of various kinds of specification logics or the check of coinductive behavioural equivalences. In this paper we develop a theory of approximation for systems of fixpoint equations in the style of abstract interpretation: a system over some concrete domain is abstracted to a system in a suitable abstract domain, with conditions ensuring that the abstract solution represents a sound/complete overapproximation of the concrete solution. Interestingly, up-to techniques, a classical approach used in coinductive settings to obtain easier or feasible proofs, can be interpreted as abstractions in a way that they naturally fit into our framework and extend to systems of equations. Additionally, relying on the approximation theory, we can characterise the solution of systems of fixpoint equations over complete lattices in terms of a suitable parity game, generalising some recent work that was restricted to continuous lattices. The game view opens the way for the development of local algorithms for characterising the solution of such equation systems. We describe a local algorithm for checking the winner on specific game positions. This corresponds to answering the associated verification question (i.e., for model checking, whether a state satisfies a formula or, for equivalence checking, whether two states are behaviourally equivalent). The algorithm can be combined with abstraction and up-to techniques, thus providing ways of speeding up the computation.

Band projections in spaces of regular operators

We introduce inner band projections in the space of regular operators on a Dedekind complete Banach lattice and study some structural properties of this class. In particular, we provide a new characterization of atomic order continuous Banach lattices as those for which all band projections in the corresponding space of regular operators are inner. We also characterize the multiplication operators L A R B L_AR_B which are band projections precisely as those with A , B A,B being band projections up to a scalar multiple.

2D Lateral Heterojunction Arrays with Tailored Interface Band Bending.

Two-dimensional (2D) lateral heterojunction arrays, characterized by well-defined electronic interfaces, hold significant promise for advancing next-generation electronic devices. Despite this potential, the efficient synthesis of high-density lateral heterojunctions with tunable interfacial band alignment remains a challenging. Here, a novel strategy is reported for the fabrication of lateral heterojunction arrays between monolayer Si2Te2 grown on Sb2Te3 (ML-Si2Te2@Sb2Te3) and one-quintuple-layer Sb2Te3 grown on monolayer Si2Te2 (1QL-Sb2Te3@ML-Si2Te2) on a p-doped Sb2Te3 substrate. The site-specific formation of numerous periodically arranged 2D ML-Si2Te2@Sb2Te3/1QL-Sb2Te3@ML-Si2Te2 lateral heterojunctions is realized solely through three epitaxial growth steps of thick-Sb2Te3, ML-Si2Te2, and 1QL-Sb2Te3 films, sequentially. More importantly, the precisely engineering of the interfacial band alignment is realized, by manipulating the substrate's p-doping effect with lateral spatial dependency, on each ML-Si2Te2@Sb2Te3/1QL-Sb2Te3@ML-Si2Te2 junction. Atomically sharp interfaces of the junctions with continuous lattices are observed by scanning tunneling microscopy. Scanning tunneling spectroscopy measurements directly reveal the tailored type-II band bending at the interface. This reported strategy opens avenues for advancing lateral epitaxy technology, facilitating practical applications of 2D in-plane heterojunctions.

Continuous lattices in formal concept analysis

Continuous lattices in formal concept analysis

Hierarchical Structures Computational Design and Digital 3D Printing

Current advances in construction automation, especially in large-scale additive manufacturing, highlight the vast potential for robots in architecture. Robotic construction is unique in its potential to reproduce highly complex structures. To advance the question of how rapid prototyping techniques are adopted in large-scale 3D printing of forms and structures, this paper presents computational methods for the design and robotic construction of cellular membranes. This research presents a comprehensive morphological model of structurally differentiated cellular membranes based on the theoretical biology model of hierarchical structures found in natural cellular solids, and, more specifically, in trabecular bone. The morphological model originates from a system of forces in equilibrium; therefore, it presents the geometric homology of a static tensional system. This research offers a methodology for the design and manufacture of meso- to large-scale triangulated geometric configurations by discrete design methods that are suitable for the robotic fused deposition of lattices and their architectural implementation in the automated manufacturing of shell structures. First, this paper explores how a form can be digitally created by geometrically emulating a given static system of forces in space. Second, inspired by the complex mechanical behavior of cancellous bone, we apply hierarchical principles found in bone remodeling to characterize discrete units that conform to continuous trabecular-like lattices. We study the geometry, limitations, opportunities for optimization, and mechanical characteristics of the lattice. The computational design methods and additive manufacturing techniques are tested in the design and construction hierarchical structures.

Commutative action logic

Abstract We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, i.e. the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic, the equational theory of *-continuous commutative action lattices. Namely, we prove that the former is $\varSigma _1^0$-complete and the latter is $\varPi _1^0$-complete. Thus, the situation is the same as in the more well-studied non-commutative case. The methods used, however, are different: we encode infinite and circular computations of counter (Minsky) machines.

Continuous [0,1]-lattices and injective [0,1]-approach spaces

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in an enriched context, where the enrichment is the quantale obtained by equipping the interval [0,1] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [0,1]-order of a separated and injective [0,1]-approach space X is a continuous [0,1]-lattice and the [0,1]-approach structure of X coincides with the Scott [0,1]-approach structure of its specialization [0,1]-order; but, unlike in the classical situation, the converse fails in general.

Unbounded continuous operators and unbounded Banach–Saks property in Banach lattices

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (uaw-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded continuous operators between Banach lattices which maps uaw-Cauchy sequences to weakly (uaw- or norm) convergent sequences. This allows us to characterize KB-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach–Saks property as an unbounded version of the weak Banach–Saks property. There are many relations between spaces possessing the unbounded Banach–Saks property with spaces fulfilled by different types of the known Banach–Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.

Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

In this article we investigate disjointly non-singular (DNS) operators. Following [9] we say that an operator T from a Banach lattice F into a Banach space E is DNS, if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [9] by showing that this class of operators forms an open subset of L(F,E) as soon as F is order continuous. Moreover, we show that in this case T is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology and the topology generated by T as a map (we say that T “complements” the unbounded norm topology in F). Since the class of DNS operators plays a similar role in the category of Banach lattices as the upper semi-Fredholm operators play in the category of Banach spaces, we investigate and indeed uncover a similar characterization of the latter class of operators, but this time they have to complement the weak topology.

Algebraic representation of frame-valued continuous lattices via the open filter monad

With a frame L as the truth value table, we prove that the collections of open filters of T0L-topological spaces form a monad. By means of L-Scott topology and the specialization L-order, we get the main results: (1) the Eilenberg-Moore algebras of the open filter monad are precisely L-continuous lattices; (2) the category of L-continuous lattices is strictly monadic over the category of T0L-topological spaces.

Strong Banach-Saks Operators

In this paper, we introduce a new class of operators, called strong Banach-Saks operators, related to the Banach-Saks and L-weakly compact operators. We first prove that every strong Banach-Saks operator from a Banach space $Z$ into a Banach lattice $F$ is Banach-Saks. Then we show that if $F$ is order continuous, the notions of strong Banach-Saks and Banach-Saks operators coincide. Finally, we close this paper by a new characterization of order continuous Banach lattices. <br><br>Вводиться новий клас операторів, так звані сильні оператори Банаха-Сакса, пов’язані з операторами Банаха-Сакса і L-слабко компактними операторами. Доведено, що кожен сильний оператор Банаха-Сакса з банахового простору $Z$ у банахову решітку $F$ є оператором Банаха-Сакса. Далі, якщо $F$ є порядково неперервним, то властивості оператора Банаха-Сакса і сильного оператора Банаха-Сакса співпадають. Нарешті, в статті дано нову характеризацію порядково неперервних банахових решіток.

Continuous Monads

Continuous monads are an axiomatic class of submonads of the double power set monad. ρ-sets are an axiomatic generalization of directed sets. The ρ-generalization of continuous lattices arises as the algebras of a continuous monad and conversely. Each ρ-continuous poset has two topologies which respectively generalize the Scott and Lawson topologies. Each ρ-contnuous lattice is compact in the canonical topology if and only if the corresponding continuous monad contains the ultrafilter monad.

A representation of continuous lattices based on closure spaces

In this paper, we establish the link between continuous lattices and closure spaces. By generalizing the notion of algebraic closure space to continuous closure space, we show that continuous lattices can be represented by continuous closure spaces, just as algebraic lattices can be represented by algebraic closure spaces. We also introduce the notion of approximable mappings between continuous closure spaces, which produces the category equivalent to that of continuous lattices with Scott-continuous functions. These results demonstrate the capability of closure spaces in representing continuous domains.

THE WAY-BELOW SOFT SET RELATION

Order is used frequently in mathematical objects and related fields. We use order in the soft set that is introduced by Molodtsov [1]. Babitha and Sunil [2] gave the definition of soft set relation, then Babitha and Sunil [3] defined partially ordered soft set. We introduced directed soft sets in [4]. Moreover, definition of soft lattice, complete soft lattice were given in [5]. We made a presentation, whose abstract was published only in the Abstract Book, in ICRAPAM 2014 [6] about the way-below soft set relation. In this paper, this study is extended and the proofs of all results of our presentation is given. Moreover, continuous soft lattices, L-soft domain are defined, and some important results are obtained and proved in this study.

Universal Approach to Z–frame Envelopes of Semilattices

In this paper we deal with systems of sets on the category of all semilattices. Based on the result of Z–frame freely generated by Z–sites studied in [D. Zhao, “Generalization of Frames and Continuous Lattices,” Ph.D. thesis, Cambridge University, 1993], we prove that every semilattice admits a Z–frame envelope. It is a universal approach for any system Z of sets. We define and study a new continuity named Za–continuty on semilattice and prove that in some specific systems of sets (Low, Fin and Idl), every Za–continuous semilattice admits a Z–continuous envelope.

The bulk-edge correspondence for continuous honeycomb lattices

We study bulk/edge aspects of continuous honeycomb lattices in a magnetic field. We compute the bulk index of Bloch eigenbundles: it equals 2 or –2, with sign depending on nearby Dirac points and on the magnetic field. We then prove the existence of two topologically protected unidirectional waves propagating along line defects. This shows the bulk/edge correspondence for our class of Hamiltonians.

Lattice-equivalence of convex spaces

The links between order and topology have been extensively studied. The objective of this paper is to study the similar links between convex spaces and order structures via the lattices of convex sets. In particular, we prove a characterization of convex spaces uniquely determined by means of their lattices of convex sets, they are precisely sober and $$S_D$$ . One adjunction between the category of convex spaces and that of continuous lattices is constructed, revealing the connection between convex structures and domain theory.

A study of the quasi covering dimension of finite lattices

The notion of dimension for posets has been studied extensively (see, for example, Bae in J Korean Math Soc 36(1):15–36, 1999, Dube et al. in Discr Math 338:1096–1110, 2015, Gierz et al. in A compendium of continuous lattices. Springer, Berlin, 1980, Trotter in Combinatorics and partially ordered sets: dimension theory. Johns Hopkins University Press, Baltimore, 1992, Vinokurov in Soviet Math Dokl 168(3):663–666, 1966, Zhang et al. in Discr Math 340(5):1086–1091, 2017). Moreover, in the view of matrix theory, some of these dimensions, such as the order dimension, the Krull dimension and the covering dimension, have been studied using the so-called order and incidence matrices (see Boyadzhiev et al. in Appl Math Comput 333:276–285, 2018, Dube et al. in Filomat 31(10):2901–2915, 2017, Georgiou et al. in Quaest Math 39(6):797–814, 2016). In this paper, we define a new dimension for finite lattices, called the quasi covering dimension, and we study many of its properties. We characterize it using matrices and we present an algorithm for computing this dimension.

Cofibrantly Generated Lax Orthogonal Factorisation Systems

The present note has three aims. First, to complement the theory of cofibrant generation of algebraic weak factorisation systems (awfss) to cover some important examples that are not locally presentable categories. Secondly, to prove that cofibrantly kz-generated awfss (a notion we define) are always lax orthogonal. Thirdly, to show that the two known methods of building lax orthogonal awfss, namely cofibrantly kz-generation and the method of “simple adjunctions”, construct different awfss. We study in some detail the example of cofibrant kz-generation that yields representable multicategories, and a counterexample to cofibrant generation provided by continuous lattices.

Fixpoint games on continuous lattices

Many analysis and verifications tasks, such as static program analyses and model-checking for temporal logics, reduce to the solution of systems of equations over suitable lattices. Inspired by recent work on lattice-theoretic progress measures, we develop a game-theoretical approach to the solution of systems of monotone equations over lattices, where for each single equation either the least or greatest solution is taken. A simple parity game, referred to as fixpoint game, is defined that provides a correct and complete characterisation of the solution of systems of equations over continuous lattices, a quite general class of lattices widely used in semantics. For powerset lattices the fixpoint game is intimately connected with classical parity games for µ-calculus model-checking, whose solution can exploit as a key tool Jurdziński’s small progress measures. We show how the notion of progress measure can be naturally generalised to fixpoint games over continuous lattices and we prove the existence of small progress measures. Our results lead to a constructive formulation of progress measures as (least) fixpoints. We refine this characterisation by introducing the notion of selection that allows one to constrain the plays in the parity game, enabling an effective (and possibly efficient) solution of the game, and thus of the associated verification problem. We also propose a logic for specifying the moves of the existential player that can be used to systematically derive simplified equations for efficiently computing progress measures. We discuss potential applications to the model-checking of latticed µ-calculi.