Prime Filters Research Articles

Spectrum Topology of a Residuated Lattice

In this paper, given a non-commutative residuated lattice , a topological space is constructed using certain fuzzy subsets of . Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice forms a topological space. Particularly, we show that this space is compact and a T0-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of is also a Hausdorff space.

A SUFFICIENT AND NECESSARY CONDITION FOR CONSTRUCTION OF 3-BAND DUAL TREE COMPLEX WAVELET

To desire expectable applications in image processing, the construction of 3-band dual tree complex wavelet is a meaningful and constructive topic. In this paper, a sufficient and necessary condition 2π/3-periodic of phase difference between the prime and dual filter banks is proposed. With this condition, it is proposed that the phase function θl(ω) of wavelet filters can be uniquely determined by the phase function θ0(ω) of the scaling filters. Imposed on a necessary constraint to θ0(ω), phase function θl(ω) can be obtained which results in the analytical of the 3-band dual tree complex wavelet. Furthermore, the complex filter banks are obtained using the wavelet matrix factorization approach. Then the dual tree 3-band complex wavelet is analytical that indicates approximately shift invariance property. In addition, it also has more direction selectivity because there are more sub-bands than dyadic wavelet transform is frequency domain.

Intuitionistic fuzzy filter theory of BL-algebras

In this paper, the intuitionistic fuzzy filter theory of BL-algebras is researched. The basic knowledge of BL-algebras and intuitionistic fuzzy sets is firstly reviewed. The notions of intuitionistic fuzzy filters, lattice filters, prime filters, Boolean filters, implicative filters, positive implicative filters, ultra filters and obstinate filters are introduced, respectively. Their important properties are investigated. In intuitionistic fuzzy sets, intuitionistic fuzzy filters, Boolean filters, ultra filters are proved to be equivalent to lattice filters, implicative filters, obstinate filters, respectively. Each intuitionistic fuzzy Boolean filter is an intuitionistic fuzzy positive implicative filter, but the converse may not be true in BL-algebras. The conditions under an intuitionistic fuzzy positive implicative filter being an intuitionistic fuzzy Boolean filter are constructed. Finally, the concepts of the intuitionistic fuzzy ultra and obstinate filters are introduced, and the intuitionistic fuzzy ultra filter is proved to be equivalent to the intuitionistic fuzzy obstinate filter in BL-algebras.

NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups Gn, r. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.

Simple characterization of strict residuated lattices with an involutive negation

We consider property of strict residuated lattices (SRL-algebras) with a new involutive negation $$\lnot, $$ called here by SRL $$_{\lnot }$$ -algebras, and give a simple characterization of SRL $$_{\lnot }$$ -algebras. We also prove a prime filter theorem of SRL $$_{\lnot }$$ -algebras, from which we conclude that every linearly ordered SRL $$_{\lnot }$$ -algebra is simple. As a corollary to this fact, we have a well-known result that every SML $$_{\lnot }$$ -algebra (SBL $$_{\lnot }$$ -algebra) is a subdirect product of linearly ordered SML $$_{\lnot }$$ -algebras (SBL $$_{\lnot }$$ -algebras).

A formal proof of the projective Eisenbud–Evans–Storch theorem

We extend a constructive proof of the Eisenbud–Evans–Storch theorem, developed in a previous work by Coquand, Schuster, and Lombardi, from the affine to the projective case. The main tool is that of distributive lattices, which allows us to replace the classical topological arguments by more algebraic and constructive ones. Given a suitable graded ring, we work in the distributive lattice in which the prime filters correspond to the homogeneous prime ideals. The proof presented here is one of the first examples of concrete results obtained using this tool.

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be the intersection of jCRL and mCRL. We prove CRL is a strict subset of biCRL which is a strict subset of both jCRL and mCRL. Let L be a DL. Then L is in mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L is in jCRL iff L has a distinguishing set of completely prime filters, and L is in CRL iff L has a distinguishing set of complete, completely prime filters. Each of the classes above is shown to be pseudo-elementary and hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.

Topological Proofs of Some Rasiowa-Sikorski Lemmas

We give topological proofs of Gornemann's adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

On the free implicative semilattice extension of a Hilbert algebra

AbstractHilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.

Fuzzy Prime Filters of Lattice Implication Algebras

In this paper, some properties of fuzzy filters are given. Besides, the structure of fuzzy filters are further studied. And finally, the concept of fuzzy prime filter is proposed with some equivalent conditions of fuzzy prime filters obtained.

Prime Filters, Normality and Irreducibility in Lattices

We recall some notions introduced and developed by Antonio Aniceto Monteiro, and show how these notions have been used and generalised, thus establishing a direct and indirect influence of Monteiro's work that extends to this day.

Topological properties of prime filters in MTL-algebras and fuzzy set representations for MTL-algebras

In the present paper, some basic properties of prime filters in MTL-algebras are studied. By introducing some topological structures on the set of all prime filters and the set of all maximal filters, respectively, and by investigating the topological properties of them, we conclude that the set of all prime filters is a compact T 0 topological space and the set of all maximal filters is a compact Hausdorff topological space. By studying the relation between valuations and filters in MTL-algebras, we prove that a MTL-algebra is homomorphic to a subalgebra of a MTL-cube over the set of all prime filters and illustrate that the Stone representation theorem of Boolean algebras is only a special case of fuzzy sets representation theorem of MTL-algebras. Thus the well-known Stone representation theorem of Boolean algebras is extended to MTL-algebras.

De Morgan Heyting algebras satisfying the identityxn(′*)≈x(n+1)(′*)

In this paper we investigate the sequence of subvarieties of De Morgan Heyting algebras characterized by the identity xn(′*) ≈ x(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in . We extend these results for finite algebras in the general case . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fuzzy Filter Spectrum of a BCK Algebra

The notion of fuzzy s‐prime filters of a bounded BCK‐algebra is introduced. We discuss the relation between fuzzy s‐prime filters and fuzzy prime filters. By the fuzzy s‐prime filters of a bounded commutative BCK‐algebra X, we establish a fuzzy topological structure on X. We prove that the set of all fuzzy s‐prime filters of a bounded commutative BCK‐algebra forms a topological space. Moreover, we show that the set of all fuzzy s‐prime filters of a bounded implicative BCK‐algebra is a Hausdorff space.

Topological aspects of the Medvedev lattice

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.

On fuzzy filters of pseudo BL-algebras

Abstract We introduce the concept of fuzzy filters of pseudo-BL-algebras and obtain some of their important properties. We further give several kinds of fuzzy filters, such as fuzzy Boolean filter, fuzzy normal filter, fuzzy prime filter, fuzzy ultrafilter and fuzzy obstinate filter and discuss the relations among the above fuzzy filters. We establish the Prime Fuzzy Filter Theorem in pseudo BL-algebras. By studying the relation between fuzzy Boolean filter and fuzzy normal filter in pseudo-BL-algebras, we solve an open problem that whether or not every Boolean filter is normal in pseudo-BL-algebras.

Pre-clinical laboratory evaluation of the new ‘AF’ arterial line filter range

The presence of emboli was recognised relatively early in the history of open heart surgery. The emboli produced during cardiopulmonary bypass have the predisposition to distribute into, and ultimately obstruct, microvessels of all tissues. The Sorin Group has recently developed a new range of arterial line filters. Before the Sorin AF range of filters was released for pre-launch clinical trials, our group performed in vitro laboratory testing of the AF range against a selection of commercially available filters on the global market. The Sorin AF620 and AF640 demonstrate both the smallest prime volume and smallest surface contact area (92ml and 290 cm(2), respectively).The results of the GME Handling Efficiency experiments ranged by 39.6%, from 95.9% to 56.3%. In terms of an air bolus handling, the results of the Limit Bolus experiment ranged by 97 ml, from 147.5 ml down to 50 ml. The pressure drop across all the filters was measured under steady state experimental conditions. All of the above investigations were considered against surface area and prime volume. It is clear from the results that some commercially available arterial line filters perform better than others, not only in overall performance, but also with regard to individual characteristics. Evaluating arterial line filters for hospital-specific use has to balance pressure drop, surface area, micro air handling, prime volume and gross air handling; all points need to be considered. In the AF620 and AF640, Sorin boast that they are the two smallest prime and smallest surface area filters commercially available on the global market. The Sorin AF filter range performs well in all of the areas we investigated and will be a competitive option for centres, irrespective of which characteristics they use to evaluate and select their arterial line filter.

Filters of residuated lattices and triangle algebras

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters. In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections. Triangle algebras characterize interval-valued residuated lattices.

Fuzzy Prime Boolean Filters and Their Operations in IMTL-algebras

Some characterizations of fuzzy prime Boolean filters of IMTL-algebras are given. The lattice operations and the order-reversing involution on the set PB(M) of all fuzzy prime Boolean filters of IMTL-algebras are defined. It is showed that the set PB(M) endowed with these operations is a complete quasi-Boolean algebra (a distributive complete lattice with an order-reversing involution). It is derived that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. By introducing an adjoint pair on PB(M), it is proved that the set PB(M) is also a residuated lattice.

TRANSITIVITY IN POINT-FREE TOPOLOGY

AbstractThe main purpose of this paper is to develop a point-free notion of topological transitivity. First, we define transitive frame maps and transitive completely prime filters inFrm, the category of frames and frame maps. Then we discuss the relationship between these notions inFrmand the notions of topological transitive and transitive points inTop. Finally, we investigate the relationship between transitive frame maps and the existence of transitive completely prime filters.