Complete Lattice Research Articles

Semiconcept and concept representations

In FCA, we often deal with a formal context K=(G,M,I) that is only partially known, i.e. only the attributes that belong to an observable set N⊂M are known. There must also exist a part H of the object set G – called a training set – that consists of elements with all attributes known. The concepts of K have to be determined using the subcontexts corresponding to the training object set H and to the observable attribute set N. In our paper, this problem is examined within the extended framework of the semiconcepts of the original context, which are generalizations of its concepts. Each semiconcept of the original context induces a semiconcept in both subcontexts. In this way, each semiconcept of the context is represented by an induced pair of semiconcepts, which can also be considered its approximations — as in the case of rough sets. We describe the properties of the mapping defined by this representation and prove that the poset formed by these semiconcept pairs is a union of two complete lattices. We show that these induced semiconcept pairs can be generated by using a simplified representation of them. As the number of semiconcepts grows exponentially with the size of the training set and the observable attribute set, an algorithm that selects the representation pairs for which their support and relevance reach a certain threshold is also presented.

Investigation on the material removal mechanism in ion implantation-assisted elliptical vibration cutting of hard and brittle material

Ductile-regime machining has been used to generate damage-free surface of hard and brittle materials by setting the cutting depth to be smaller than the ductile-brittle transition depth (DBTD). However, the ductile-regime cutting of sapphire remains challenging owing to its extreme hardness, small DBTD, serious surface fractures, and severe tool wear. To solve this problem, ion implantation-assisted elliptical vibration cutting (Ii-EVC) has been proposed in this study to enhance the machinability of hard and brittle materials. Taking sapphire as an example, high-energy phosphorus ions were implanted into the workpiece to modify its surface. Nanoindentation tests revealed that the modified materials undergo plastic and elastic deformation more easily due to the decrease in hardness and modulus. Compared with nanocutting without implantation assistance, the DBTD of implanted sapphire has been increased by more than five times. The advantageous effects of Ii-EVC achieve great enhancement in machinability, including surface fractures suppression, tool-wear reduction, chips morphology transformation from discontinuous to continuous, and cutting force decrease. Furthermore, even near the cracks in the brittle region after Ii-EVC, the subsurface microstructure showed a more complete lattice arrangement and a strain distribution close to zero, indicating that crack propagation was effectively suppressed. Due to the promoted localized plastic deformation, the stress distribution in the implanted material is much smaller than that in pristine workpiece. Implantation-induced defects not only serve as a core for absorbing external energy from the high-frequency vibration and improving the in-grain deformation but also facilitate the formation of shear bands. The interface with high distortion between the modified layer and substrate can effectively dissipate strain energy and hinder crack propagation to the free surface. The turning experiments verified that Ii-EVC can achieve better surface quality, less tool wear and higher optical transmittance. Overall, Ii-EVC addresses the challenges of tool breakage and surface fracture caused by high-frequency collision between tool and workpiece in traditional EVC, overcomes the problem of limited modification depth in ion implantation, and increases the ductile-regime removal depth of extremely hard and brittle materials to several microns. Such findings demonstrate that Ii-EVC is a promising method for the ultra-precision manufacturing of advanced materials.

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.

Antitone involutions on tensor products of complete lattices and complete distributivity

The purpose of this paper is to study antitone involutions on tensor products of complete lattices. A lattice with antitone involution is called an involution lattice. We show that if M is a completely distributive involution lattice, then for each complete involution lattice L there exists a unique antitone involution on the tensor product M⊗L such that the natural embeddings of M and L into M⊗L are involution-preserving. This is best possible, since the described property characterizes complete distributivity in the class of complete involution lattices. When M and L are completely distributive involution lattices, M⊗L with the aforementioned antitone involution is the codomain of a universal bimorphism in the sense of the category of all completely distributive de Morgan algebras and their join- and involution-preserving maps. The case that M and L are orthocomplemented is explored too.

A new minimal element theorem and new generalizations of Ekeland’s variational principle in complete lattice optimization problem

A new minimal element theorem and new generalizations of Ekeland’s variational principle in complete lattice optimization problem

One-point connectifications of regular spaces

It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the Alexandroff compactification) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property P and give conditions on P which guarantee the space to have a regular one-point connectification with P.

Generalization of filter theory of EQ-algebras based on lattices

Abstract In this article, considering the notion of a lattice-valued set (${\mathcal{L}}$-set, for short) we introduce the notions of ${\mathcal{L}}$-prefilter and ${\mathcal{L}}$-filter in EQ-algebras. We provide several characterizations and equivalent conditions for these concepts and also characterize the ${\mathcal{L}}$-prefilter and ${\mathcal{L}}$-filter generated by an ${\mathcal{L}}$-set. Subsequently, we study the lattice structure of these filters and prove that in an $\ell $EQ-algebra, the lattice of ${\mathcal{L}}$-prefilters is a complete Brouwerian lattice, and hence it forms a Heyting lattice. In a residuated $\ell $EQ-algebra, we show that the lattice of ${\mathcal{L}}$-filters is also a Heyting lattice. They also form a semi-De Morgan algebra. Moreover, we demonstrate that the skeleton of an $\ell $EQ-algebra under appropriate operations forms a Boolean algebra. Furthermore, we introduce (relative) ${\mathcal{L}}$-congruences in EQ-algebras and investigate their properties. We also explore the relationships between ${\mathcal{L}}$-prefilters/${\mathcal{L}}$-filters and ${\mathcal{L}}$-congruences. We prove that ${\mathcal{L}}$-prefilters induce a relative ${\mathcal{L}}$-equivalence relation and ${\mathcal{L}}$-filters correspond to relative ${\mathcal{L}}$-congruences, and state and prove some isomorphism theorems.

Three-way concept lattice from adjunctive positive and negative concepts

Three-way concept lattices (TCLs) have been widely explored due to their clear hierarchical structures, concise visual description and good interpretability. In contrast to classic formal contexts, lattice-valued fuzzy contexts exhibit great capability in describing and representing concepts with uncertainty. Different from conventional approaches to research of TCLs, this paper focuses on investigating the algebraic structure and properties of three-way concept lattice (TCL) stemmed from the positive concept lattice and negative concept lattice in a lattice-valued formal context. Several associated concept lattices such as the Cartesian product of positive concept lattice and negative lattice (i.e., pos-neg lattice), lattices induced from the partition of the pos-neg lattice, and their relationship are explored. Specifically, the isomorphism, embedding and order-preserving mappings between them are built. The quotient set of pos-neg lattice when being defined a specific equivalence relation on it is a complete lattice and each equivalence class is a lower semi-lattice. It is further declared that the structure of TCL is intrinsically and determined wholly by the pos-neg lattice. A practical application of the built theory of TCL is provided to sort alternatives in multi-criteria decision making.

O-fuzzy ideals of [formula omitted]-valued general fuzzy automata

This study aims to examine the concept of O-fuzzy ideals of “LB-valued general fuzzy automata” (for simplicity, LB-valued GFA), where B is regarded as a set of propositions about the GFA, and its underlying structure is a complete distributive lattice as a generalization of δ-fuzzy ideals. We provide a necessary and sufficient condition for χ{0} to be an O-fuzzy ideal, and we investigate the relationship between the concepts of O-fuzzy ideals and α-fuzzy ideals in a set B. Accordingly, the related notions and the results obtained in this study are clarified and explicated by some examples.

Polynomial crisp-minimization algorithm for fuzzy deterministic automata

Designing automata minimization algorithms is a significant topic in Automata Theory and Languages with practical applications. In this paper, we develop an efficient minimization algorithm for deterministic fuzzy finite automata over locally finite lattices. More precisely, the algorithm outputs an equivalent minimal crisp-deterministic fuzzy finite automaton for an input fuzzy deterministic finite automaton (FDfA). The running time of the proposed algorithm is polynomial for particular types of locally finite lattices, specifically for max-min-based complete residuated lattices. The intuition behind the proposed algorithm relies on the polynomial minimization algorithm's intuition for ordinary deterministic automata developed by Vazquez de Parga, Garcia, and Lopez (2013) [35]. The motivation for this study comes from the fact that the original algorithm's notions and meanings are lost in the context of fuzzy automata. Thus, we establish a new theoretical foundation that provides the correctness and the polynomial-time nature of this new crisp-minimization algorithm for FDfAs.

Two-mode weakly linear systems of fuzzy relation equations: Structures of solutions, computation methods, and applications

The problem of aggregating large fuzzy relational structures is of major importance. One of the main issues is seeking a simpler fuzzy relational structure, while keeping the structure properties of the original fuzzy relational structure as much as possible. This paper covers the mathematical foundations of the described problem, and deals with the solvability issues of two-mode weakly linear systems over complete residuated lattices. Within these systems, our objective is to identify a pair of unknown fuzzy relations that are present on both sides of the equation or inequality. We provide a characterization of the set of all pairs of fuzzy relations that are solutions to two-mode weakly linear systems to a desired degree, and develop an algorithm to compute the greatest such pair. We also present an algorithm to determine a pair that solves a two-mode weakly linear system to a given extent, such that each component is a fuzzy preorder (resp. fuzzy equivalence). Ultimately, we demonstrate the application aspects of aggregating fuzzy relational structures.

Orthogonally additive operators on complex vector lattices

In this article we continue investigation of the lateral order on complex vector lattices started in [10,39]. We extend some of main results of [35] and [32] to the setting of operators on complex vector lattices. We establish the Riesz-Kantorovich type calculus for regular orthogonally additive operators defined on the complexification EC of an uniformly complete vector lattice E with the principal projection property and taking values in Dedekind complete vector lattice F. We prove that a regular orthogonally additive operator T:EC→F from the complexfication EC of an uniformly complete vector lattice E with the principal projection property to a Banach lattice F with an order continuous norm is narrow if and only if the modulus |T|:EC→F is.

F-transforms determined by overlap and grouping maps over a complete lattice

F-transforms determined by overlap and grouping maps over a complete lattice

Operations on Fixpoint Equation Systems

We study operations on fixpoint equation systems (FES) over arbitrary complete lattices. We investigate under which conditions these operations, such as substituting variables by their definition, and swapping the ordering of equations, preserve the solution of a FES. We provide rigorous, computer-checked proofs. Along the way, we list a number of known and new identities and inequalities on extremal fixpoints in complete lattices.

Differentiable, holomorphic, and analytic functions on complex Φ-algebras

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex Φ-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.

Free uniformly complete vector lattices

Free uniformly complete vector lattices

On Extreme Extension of Positive Operators

Given vector lattices $E$, $F$ and a positive operator $S$ from a majorzing subspace $D$ of $E$ to~$F$, denote by $\mathcal{E}(S)$ the collection of all positive extension of $S$ to all of $E$. This note aims to describe the collection of extreme points of the convex set $\mathcal{E}(T\circ S)$. It is proved, in particular, that $\mathcal{E}(T\circ S)$ and $T\circ\mathcal{E}(S)$ coincide and every extreme point of $\mathcal{E}(T\circ S)$ is an~extreme point of $T\circ\mathcal{E}(S)$, whenever $T:F\to G$ is a~Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in~Kantorovich spaces, and an intrinsic characterization of subdifferentials.

Lift and generalized ordinal sum of negations on bounded posets

In this paper, two construction methods for fuzzy negations on bounded structures are presented. The first method focuses on lifting a fuzzy negation from a more general subposet to the complete lattice, extending the existing research. It provides an effective approach to lift fuzzy negations, even when the top and bottom elements of the subposet differ from those of the complete lattice. The second method explores the generalized ordinal sum of negations on an abelian partially ordered group, achieved by complementing a given fuzzy negation. A sufficient and necessary condition for the generalized ordinal sum, with a familiar fuzzy negation as the complement, to be a fuzzy negation is given. Building upon this, the paper introduces a supplementary fuzzy negation based on the provided family of subintervals, which guarantees that the generalized ordinal sum is still a fuzzy negation.

A representation of sup-completion

It was showed by Donner in [Extension of positive operators and Korovkin theorems, Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X X may be embedded into a cone X s X^s , called the sup-completion of X X . We show that if one represents the universal completion of X X as C ∞ ( K ) C^\infty (K) , then X s X^s is the set of all continuous functions from K K to [ − ∞ , ∞ ] [-\infty ,\infty ] that dominate some element of X X . This provides a functional representation of X s X^s , as well as an easy alternative proof of its existence.

Progress toward the Definition of X-ray Computed Tomography Accuracy in the Characterization of Polymer-Based Lattice Structures.

Lattice structures have become an innovative solution for the improvement of part design, as they are able to substitute solid regions, maintain mechanical capabilities, and reduce material usage; however, dimensional quality control of these geometries is challenging. X-ray computed tomography (XCT) is the most suitable non-destructive metrological technique as it is capable of characterizing internal features and hidden elements. Uncertainty estimation of XCT is still in development, and studies typically use high-resolution calibrated devices such as focal variation microscopes (FVMs) as a reference, focusing on certain parts of the lattice but not the whole structure. In this paper, an estimation of the accuracy of XCT evaluation of a complete lattice structure in comparison to a higher-resolution reference device (FVM) is presented. Experimental measurements are taken on ad hoc designed test objects manufactured in polyamide 12 (PA12) using selective laser sintering (SLS), optimized for the evaluation on both instruments using different cubic-based lattice typologies. The results confirm higher precision on XCT evaluation in both qualitative and quantitative analysis. Even with a lower resolution, XCT is able to characterize details of the surface such as re-entrant features; as well, standard deviations and uncertainties in strut diameter evaluation remain more stable in all cells in XCT, identifying on the other hand reconstruction problems on FVM measurements. Moreover, it is shown that, using XCT, no additional evaluation errors were found in inner cells, suggesting that the measurement of external elements could be representative of the whole structure for metrological purposes.