Principal Filters Research Articles

On distributive lattices of simple ordered semirings

In this paper, we study principal ordered ideals of ordered semirings, and give their descriptions with some properties. The characterization of simple ordered semirings is studied, and a relation between a prime ordered ideal and filter has been developed. Finally, we characterize the ordered semirings which are distributive lattices of simple ordered semirings. Here, we find that the equivalence relations [Formula: see text] and [Formula: see text] induced, respectively, from principal filters and principal ordered ideals coincide as distributive lattice congruence in an ordered semiring [Formula: see text] whenever [Formula: see text] is a distributive lattice of simple ordered semirings.

Subobjects and Compactness in Point-Free Convergence

We consider subobjects in the context of point-free convergence (in the sense of Goubault-Larrecq and Mynard), characterizing extremal monomorphisms in the opposite category of that of convergence lattices. It turns out that special ones are needed to capture the notion of subspace. We call them standard and they essentially depend on one element of the convergence lattice. We introduce notions of compactness and closedness for general filters on a convergence lattice, obtaining adequate notions for standard extremal monos by restricting ourselves to principal filters. The classical facts that a closed subset of a compact space is compact and that a compact subspace of a Hausdorff space is closed find generalizations in the point-free setting under the form of general statements about filters. We also give a point-free analog of the classical fact that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism.

Join-semilattices whose principal filters are pseudocomplemented lattices

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a∨b in the section [b,1] is denoted by a → b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with the so-called non-classical implication semilattices which can be converted into I-algebras having everywhere defined operations. Moreover, we relate our structures to sectionally and relatively residuated semilattices which means that our logical structures are closely connected with substructural logics. We show that I-algebras form a congruence distributive, 3-permutable and weakly regular variety.

The contribution of GIS and remote Sensing for mapping fracture networks in the northern border western High Atlas Mafrrakech (Morocco)

The main objective of this work is to test the use of Landsat Oli Tirs imagery for structural identification in the region of the Aït Ourir basins (High Western Atlas). Different processing techniques were used (color composites, band ratios, principal component analysis and directional filters) which provide an excellent discrimination between the different lithological formations and helped us to establish lineament mapping. The results were validated with the fieldwork and bibliography that are essential in any study based on remote sensing. Finally, a lithostructural map of the study area was established on the basis of the results obtained in this work.

Generalized prime $D$-filters of distributive lattices

The concept of generalized prime $D$-filters is introduced in distributive lattices. Generalized prime $D$-filters are characterized in terms of principal filters and ideals. The notion of generalized minimal prime $D$-filters is introduced in distributive lattices and properties of minimal prime $D$-filters are then studied with respect to congruences. Some topological properties of the space of all prime $D$-filters of a distributive lattice are also studied.

1-Safe Petri Nets and Special Cube Complexes

Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By a result of Thiagarajan [46], these unfoldings are exactly the trace-regular event structures. Thiagarajan [46] conjectured that regular event structures correspond exactly to trace-regular event structures. In a recent paper (Chalopin and Chepoi [12]), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. However, we proved that Thiagarajan’s conjecture is true for regular event structures whose domains are principal filters of universal covers of finite special cube complexes. In the current article, we prove the converse: To any finite 1-safe Petri net N , one can associate a finite special cube complex X N such that the domain of the event structure E N (obtained as the unfolding of N ) is a principal filter of the universal cover X̃ N of X N . This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace-regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity), we disprove yet another conjecture by Thiagarajan (from the paper with Yang [48]) that the monadic second-order logic of a 1-safe Petri net (i.e., of its event structure unfolding) is decidable if and only if its unfolding is grid-free. It was proven by Thiagarajan and Yang [48] that the MSO logic is undecidable if the unfolding is not grid-free. Our counterexample is the trace-regular event structure that arises from a virtually special square complex Z. The domain of this event structure Ė Z is the principal filter of the universal cover Z̃ of Z in which to each vertex we added a pendant edge. The graph of the domain of Ė Z has bounded hyperbolicity (and, thus, the event structure Ė Z is grid-free) but has infinite treewidth. Using results of Seese, Courcelle, and Müller and Schupp, we show that this implies that the MSO theory of the event structure Ė Z is undecidable.

The monoid of order isomorphisms between principal filters of $${\varvec{\mathbb {N}}}^n$$Nn

Let n be any positive integer and be the semigroup of all order isomorphisms between principal filters of the nth power of the set of positive integers $${\mathbb {N}}$$ with the product order. We study algebraic properties of the semigroup . In particular, we show that is a bisimple, E-unitary, F-inverse semigroup, describe Green’s relations on and its maximal subgroups. We show that the semigroup is isomorphic to the semidirect product of the direct nth power of the bicyclic monoid by the group of permutation . Also we prove that every non-identity congruence $${\mathfrak {C}}$$ on the semigroup is a group and describe the least group congruence on . We show that every Hausdorff shift-continuous topology on is discrete and discuss embedding of the semigroup into compact-like topological semigroups.

What are the principal environmental filters driving species composition and succession on mineralogically different spoil heaps?

ABSTRACTThe relationship between selected environmental variables and plant species composition was studied on two mineralogically different spoil heaps (Hg and Cu) in Central Slovakia with contrasting reclamation approaches. Data on plant species composition were collected by stratified random sampling in defined physiognomic vegetation types. A detrended correspondence analysis showed that most of the variability in species composition was related to the succession gradient from open communities with a low cover of vascular plants to forest vegetation, and to the moisture gradient. Variance partitioning by canonical correspondence analysis revealed that most of the variability in plant composition was related to the content of various heavy metals (27.8% at the Hg-spoil heap and 28.3% at the Cu-spoil heap), but a significant relationship was found only for Mn. Other significant factors comprised soil moisture, pH and P content for the Hg-spoil heap and soil temperature and Ca content for the Cu-spoil heap. Although heavy metal content explained most of the variability in species composition, the relationship was caused by the correlation of heavy metal content with other environmental variables rather than by a direct causal relationship.

A PCA–CCA network for RGB-D object recognition

Object recognition is one of the essential issues in computer vision and robotics. Recently, deep learning methods have achieved excellent performance in red-green-blue (RGB) object recognition. However, the introduction of depth information presents a new challenge: How can we exploit this RGB-D data to characterize an object more adequately? In this article, we propose a principal component analysis–canonical correlation analysis network for RGB-D object recognition. In this new method, two stages of cascaded filter layers are constructed and followed by binary hashing and block histograms. In the first layer, the network separately learns principal component analysis filters for RGB and depth. Then, in the second layer, canonical correlation analysis filters are learned jointly using the two modalities. In this way, the different characteristics of the RGB and depth modalities are considered by our network as well as the characteristics of the correlation between the two modalities. Experimental results on the most widely used RGB-D object data set show that the proposed method achieves an accuracy which is comparable to state-of-the-art methods. Moreover, our method has a simpler structure and is efficient even without graphics processing unit acceleration.

The Fuzzy Lattice of Ideals and Filters of an Almost Distributive Fuzzy Lattice

In this paper, the concept of fuzzy lattice is discussed. It is proved that a fuzzy poset (IA(L),B) and (FA(L),B) forms a fuzzy lattice, where IA(L) and FA(L) are the set containing all ideals, and the set containing all filters of an Almost Distributive Fuzzy Lattice(ADFL) respectively. In addition we proved that, a fuzzy poset (PIA(L),B) and (PFA(L),B) forms fuzzy distributive lattice, where PIA and PFA(L) denotes the set containing all principal ideals and the set containing all principal filters of an ADFL. Finally, it is proved that for any ideal I and filter F of an ADFL, IiA = {(i]A : i in I} and FfA = {[f)A : f in F} are ideals of a fuzzy distributive lattice (PIA(L),B) and (PFA(L),B) respectively, and FiA = {(f]A : f in F} and IfA = {[i)A : i in I} are filters of a distributive fuzzy lattice (PIA(L),B) and (PFA(L),B) respectively.

Orbits of Maximal Vector Spaces

Let V ∞ be a standard computable infinite-dimensional vector space over the field of rationals. The lattice $$ \mathfrak{L} $$ (V ∞ ) of computably enumerable vector subspaces of V ∞ and its quotient lattice modulo finite dimension, $$ \mathfrak{L} $$ *(V ∞ ), have been studied extensively. At the same time, many important questions still remain open. In 1998, R. Downey and J. Remmel posed the question of finding meaningful orbits in $$ \mathfrak{L} $$ *(V ∞ ) [4, Question 5.8]. This question is important and difficult and its answer depends on significant progress in the structure theory for the lattice $$ \mathfrak{L} $$ *(V ∞ ), and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit of $$ \mathfrak{L} $$ *(V ∞ ). More specifically, we consider two vector spaces, V 1 and V 2 , which are spanned by two quasimaximal subsets of, possibly different, computable bases of V ∞ . We give a necessary and sufficient condition for the principal filters determined by V 1 and V 2 in $$ \mathfrak{L} $$ *(V ∞ ) to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism Φ of $$ \mathfrak{L} $$ *(V ∞ ) such that Φ maps the equivalence class of V 1 to the equivalence class of V 2 . Our results are expressed using m-degrees of relevant sets of vectors. This study parallels the study of orbits of quasimaximal sets in the lattice e of computably enumerable sets, as well as in its quotient lattice modulo finite sets, e * , carried out by R. Soare in [13]. However, our conclusions and proof machinery are quite different from Soare’s. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space in $$ \mathfrak{L} $$ *(V ∞ ) is generally much more complicated than the one of a principal filter determined by a quasimaximal set in e* . We also state that, unlike in e* , having isomorphic principal filters in $$ \mathfrak{L} $$ *(V ∞ ) is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of $$ \mathfrak{L} $$ *(V ∞ ).

잡음 밀도에 따라 가변 마스크를 적용한 Salt and Pepper 잡음 제거에 관한 연구

In digital era image processing has been utilized in a variety of media such as TV, camera and smart phone. Typically salt and pepper noise are generated by various causes during the analysis, identification, and processing of image data. Principal filters such as SMF, CWMF, and AMF have been used to remove these noise. But the existing filters fall short of edge preservation and noise elimination in high noise densities. Thus, a processing algorithm, on which the size of deformable mask varies depending on the noise density, is proposed to remove salt and pepper noise effectively in this study. The performance of the proposed method was evaluated compared with the existing methods using PSNR.

Skew residuated lattices

We replace the so-called adjointness in the definition of residuated lattice by its strict version where inequalities are replaced by equalities. We prove that such structures, called skew residuated lattices, can be characterized as lattices with certain involutions in principal filters. Since skew residuated lattices have the cancellation property, they are close to divisibility loops introduced by B. Bosbach in 1988. We show under what condition can skew residuated lattices be represented by such loops.

Intuitionistic logic and Muchnik degrees

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now clas- sic result of Skvortsova for the Medvedev lattice. 1. Introduction Among the structures arising from computability theory, the lattices intro- duced by Medvedev and Muchnik stand out for several distinguished features and a broad range of applications. In particular these lattices have additional structure that makes them suitable as models of certain propositional cal- culi. The structure of the Medvedev lattice as a Brouwer algebra, and thus as a model for propositional logics, has been extensively studied in several papers, see e.g. (10), (15), (17), (20), (22). Originally motivated in (10) as a formalization of Kolmogorov's calculus of problems (7), the Medvedev lattice fails to provide an exact interpretation of the intuitionistic propositional cal- culus IPC; however, as shown by Skvortsova (15), there are initial segments of the Medvedev lattice that model exactly IPC. On the other hand, little is known about the structure of the Muchnik lattice, and of its dual, as Brouwer algebras. The goal of this paper is to show that there are initial segments (equivalently: factors obtained dividing the lattice by principal filters) of the Muchnik lattice, in which the set of valid propositional sentences coincides with IPC. This shows that the analogue of Skvortsova's theorem also holds for the Muchnik lattice. From this, it readily follows that the propositional sentences that are valid in the Muchnik lattice are exactly the sentences of the so-called logic of the weak law of the excluded middle ((17)). Similar results (as announced, with outlined proofs, in (18)) hold of the dual of the Muchnik lattice: detailed proofs are provided in Section 5. For all unexplained notions from computability theory, the reader is re- ferred to Rogers (14); our main source for Brouwer algebras and the algebraic semantics of propositional calculi is Rasiowa-Sikorski (13). A comprehensive survey on the Medvedev and Muchnik lattices, and their mutual relation- ships, can be found in (19). Throughout the paper we use the symbols + and × to denote the join and meet operations, respectively, in any lattice. 1.1. The Medvedev and the Muchnik lattices. Although our main ob- ject of study is the Muchnik lattice, reference to the Medvedev lattice will be sometimes useful. Therefore, we start by reviewing some basic definitions and facts concerning both lattices. Following Medvedev (10), a mass problem is a set of functions from the set of natural numbers ω to ω. There are two natural ways to extend Turing reducibility to mass problems: following (10),

On Double Basic Algebras and Pseudo-effect Algebras

Double basic algebras are a counterpart of bounded lattices with order-antiautomorphisms on principal filters. In the paper, an independent axiomatization of double basic algebras is given and lattice pseudo-effect algebras are characterized in the setting of double basic algebras.

Which O-commutative Basic Algebras Are Effect Algebras

By a basic algebra is meant an MV-like algebra (A,⊕,¬,0) of type 〈2,1,0〉 derived in a natural way from bounded lattices having antitone involutions on their principal filters. We show that (i) atomic Archimedean basic algebras for which the operation ⊕ is o-commutative are effect algebras and (ii) atomic Archimedean commutative basic algebras are MV-algebras. This generalizes the results by Botur and Halas on finite commutative basic algebras and complete commutative basic algebras.

Modulated logics and flexible reasoning

This paper studies a family of monotonic extensions of first-order logic which we call modulated logics, constructed by extending classical logic through generalized quantifiers called modulated quantifiers. This approach offers a new regard to what we call flexible reasoning. A uniform treatment of modulated logics is given here, obtaining some general results in model theory. Besides reviewing the “Logic of Ultrafilters”, which formalizes inductive assertions of the kind “almost all”, two new monotonic logical systems are proposed here, the “Logic of Many” and the “Logic of Plausibility”, that characterize assertions of the kind “many”, and “for a good number of”. Although the notion of simple majority (“more than half”) can be captured by means of a modulated quantifier semantically interpreted by cardinal measure on evidence sets, it is proven that this system, although sound, cannot be complete if checked against the intended model. This justifies the interest on a purely qualitative approach to this kind of quantification, what is guaranteed by interpreting the modulated quantifiers as notions of families of principal filters and reduced topologies, respectively. We prove that both systems are conservative extensions of classical logic that preserve important properties, such as soundness and completeness. Some additional perspectives connecting our approach to flexible reasoning through modulated logics to epistemology and social choice theory are also discussed.

A class of $${\Sigma _{3}^{0}}$$ modular lattices embeddable as principal filters in $${\mathcal{L}^{\ast }(V_{\infty })}$$

Let I 0 be a a computable basis of the fully effective vector space V ∞ over the computable field F. Let I be a quasimaximal subset of I 0 that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter $${\mathcal{L}^{\ast}(V,\uparrow )}$$ of V = cl(I) is isomorphic to the lattice $${\mathcal{L}(n, \overline{F})}$$ of subspaces of an n-dimensional space over $${\overline{F}}$$ , a $${\Sigma _{3}^{0}}$$ extension of F. As a corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices $${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$$ is isomorphic to a principal filter of $${\mathcal{ L}^{\ast}(V_{\infty})}$$ . We thus answer Question 5.3 “What are the principal filters of $${\mathcal{L}^{\ast}(V_{\infty}) ?}$$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets.

Weak MV-algebras

Abstract In a recent paper [CHAJDA, I.—KÜHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property.

On a general form of join matrices associated with incidence functions

Let \((P,\leq) = (P, \vee)\) be a join-semilattice with finite principal order filters and let \(\Psi_{\vee}\) denote the function on P × P defined by $$\Psi_{\vee}(x, y) = \sum\limits_{x\vee y\leq z} f(x, z)g(y,z)h(z),$$ where f and g are incidence functions of P and h is a complex-valued function on P. We calculate the determinant and the inverse of the matrix \([\Psi_{\vee}(x_i, x_j)]\), where S = {x1, x2,...,xn} is a join-closed subset of P and (xi, xj ∈S, xi ≤ xj ≤ z) ⇒ f(xi, z) = f(xj , z) holds for all z ∈ P.