Complete Lattice Research Articles

Extensible Proof Systems for Infinite-State Systems

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

On $$(G_O,O)-$$fuzzy rough sets based on overlap and grouping functions over complete lattices

On $$(G_O,O)-$$fuzzy rough sets based on overlap and grouping functions over complete lattices

Some properties of conjunctivity (subfitness) in generalized settings

The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T 1-separation, appeared (as disjunctivity) already in the pioneering Wallman’s [16], then practically disappeared to reappear again (conjunctivity, subfitness), until it was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss it on the background of the systems of meet-sets (subsets closed under existing infima) in posets of various generality (semilattices, lattices, distributive lattices, complete lattices) and present parallels of some localic (frame) facts, including a generalized variant of fitness.

The geometry of discrete L-algebras

Abstract The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined.

Convex Monotone Semigroups on Lattices of Continuous Functions

We consider convex monotone C_0 -semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a \sigma -Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton–Jacobi–Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow us to define the generators in a weak sense.

Mid-infrared modulated photoluminescence mapping to investigate in-plane distributions of bandedge transitions in As-doped HgCdTe

In-plane distribution of band edge electronic structure of the narrow-gap arsenic (As)-doped HgCdTe is a crucial topic fundamentally and technically for mid-infrared detector array but hard to be investigated for long experimentally. This work reports a demonstration of the modulated photoluminescence (PL) mapping carried on As-doped HgCdTe, with the spatial resolution of a typical array pixel scale and with the elaborate band edge transitions being distinguished. Curve fittings related to the bandgap and to the impurities of Hg vacancy (VHg), As donor (AsHg), and AsHg–VHg pair are confirmed for PL mapping assisted by a temperature-dependent PL analysis. The spatial non-uniformity of Cd composition and the distributions of impurity relative concentrations are derived. Correlation analysis suggests that the As atom directly replacing the Hg atom in the complete lattice and the As atom occupying the already-existing VHg are the two approaches for the AsHg formation during As doping. The non-uniform band edge properties may lead to the in-plane conductivity fluctuation detrimental to the array performance. The results indicate the significance of probing the band edge impurity uniformity of As-doped HgCdTe for arrays fabrication, and the modulated PL mapping serving as a desirable and effective tool.

Multiplicative lattices, idempotent endomorphisms, and left skew braces

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure [Formula: see text], it is possible to derive several notions concerning [Formula: see text] in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of [Formula: see text] with multiplication given by commutator of congruences. Our main application is to the study of some notions concerning left skew braces.

A study of complete lattices of covering spaces

Let C(X) denote the set of all covering spaces (X', x', p) of (X,x) where (X,x) are path connected,locally path connected and semilocally simply connected pointed topological spaces. In [1], it is shown that (C(X),≥) is a lattice with respect to the partial order relation ≥. Also (C(X),≥) is a modular, bounded and complete lattice when π(X,x) is abelian. In this paper, we will study some properties of the complete lattice (C(X), ≥).

On the lattice of conatural classes of linear modular lattices

The collection of all cohereditary classes of modules over a ring R is a pseudocomplemented complete big lattice. The elements of its skeleton are the conatural classes of R-modules. In this paper we extend some results about cohereditary classes in R-Mod to the category mathcal {L_{M}} of linear modular lattices, which has as objects all complete modular lattices and as morphisms all linear morphisms. We introduce the big lattice of conatural classes in mathcal {L_{M}}, and we obtain some results about it, paralleling the case of R-Mod and arriving at its being boolean. Finally, we prove some closure properties of conatural classes in mathcal {L_{M}}.

Limits of vector lattices

If K is a compact Hausdorff space so that the Banach lattice C(K) is isometrically lattice isomorphic to a dual of some Banach lattice, then C(K) can be decomposed as the ℓ∞-direct sum of the carriers of a maximal singular family of order continuous functionals on C(K). In order to generalise this result to the vector lattice C(X) of continuous, real valued functions on a realcompact space X, we consider direct and inverse limits in suitable categories of vector lattices. We develop a duality theory for such limits and apply this theory to show that C(X) is lattice isomorphic to the order dual of some vector lattice F if and only if C(X) can be decomposed as the inverse limit of the carriers of all order continuous functionals on C(X). In fact, we obtain a more general result: A Dedekind complete vector lattice E is perfect if and only if it is lattice isomorphic to the inverse limit of the carriers of a suitable family of order continuous functionals on E. A number of other applications are presented, including a decomposition theorem for order dual spaces in terms of spaces of Radon measures.

Composition series of arbitrary cardinality in modular lattices and abelian categories

For a certain family of complete modular lattices, we prove a “Jordan–Hölder–Schreier-like” theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices with are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5⁎). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.

Fuzzy concept lattices induced by doubly distance spaces

We introduce three algebraic structures induced by a doubly distance space on a complete co-residuated lattice. We show that (1) the two families of f-stable fuzzy sets and g-stable fuzzy sets induced by a doubly distance space are complete lattices and (2) these two families are isomorphic as lattices. We define the fuzzy concept lattice induced by a doubly distance space and show that this fuzzy concept lattice forms a complete lattice. Moreover, we give two examples related to them for an information system.

L-fuzzy filters on complete residuated lattices

This paper is toward the establishment of relationships between L-fuzzy filters, L-fuzzy topological spaces and L-fuzzy pre-proximity spaces in complete residuated lattices. We have demonstrated the existence of functors between the categories of L-fuzzy filter spaces, L-fuzzy topological spaces and L-fuzzy pre-proximity spaces.

Morpho-logic from a topos perspective – application to symbolic AI

Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of a topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

On torsion theories and open classes of linear modular lattices

Given a ring R, a bijection exists between torsion theories and idempotent radicals. Further, the class of hereditary torsion theories contains the skeleton of the open classes. In this work, we extend the notions and main results in R-Mod about torsion classes, torsion-free classes, and open classes to the category of linear lattices, whose objects are complete modular lattices and whose morphisms are linear morphisms.

Not every countable complete distributive lattice is sober

AbstractThe study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $\Sigma P\times \Sigma Q$ coincides with the Scott topology of the product poset $P\times Q$ if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and $\Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.

Representation theorems for regular operators

AbstractWe elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation , where is the sum of an absolutely order summable family of disjointness preserving operators and is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators.

Adjunctive three-way concepts from positive and negative concepts in lattice-valued formal contexts

Three-way concept analysis is an integration of formal concept analysis and three-way decision, which has been found wide applications in knowledge representation, cognitive computing and decision sciences. There have been already a huge number of literature on theory and applications on such topics. Most researches on formal concept analysis focus, independently, on specific scenarios such as Boolean formal contexts and fuzzy formal contexts. In this paper, a unified framework for three-way concept analysis is built in lattice-valued formal contexts. The notions of positive concept and negative concept are given in a lattice-valued formal context based on an overlapping function, one of significant operators for information fusion. Two pairs of object induced adjunctive three-way concept operators and attribute induced adjunctive three-way concept operators are introduced. Their properties have been fully investigated. With those operators, the algebraic structures of adjunctive three-way concepts, their extensions and intensions have been explored. Specifically, the local structure within pairs of sub-concept and sup-concept has been finely characterized. The derived results provide ones a way to construct the complete lattice of a lattice-valued formal context or its sublattice by determining all pairs of sub-concept and sup-concept provided that an object fuzzy set or an attribute fuzzy set is given. A case analysis on a real world dataset from UCI repository is presented to verify the rationality and significance of obtained results.

Lateral order on complex vector lattices and narrow operators

AbstractIn this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

Fuzzy Initial and Final Segments in ADL’s

In this paper, we define the concepts of fuzzy initial and final segments in an Almost Distributive Lattice (ADL) and certain properties of these are discussed. It is proved that the set of fuzzy initial segments forms a complete lattice and that the set of fuzzy final segments of an ADL A forms a complete lattice if and only if A is a bounded distributive lattice.