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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A note on good pseudo BL-algebras

100%
Wojciechowska-Rysiawa M.
Discussiones Mathematicae - General Algebra and Applications
|
2010
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tom 30
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nr 2
193-205
EN
Pseudo BL-algebras are a noncommutative extention of BL-algebras. In this paper we study good pseudo BL-algebras and consider some classes of these algebras.
2
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Strong meager properties for filters

100%
Laflamme C.
Fundamenta Mathematicae
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1994-1995
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tom 146
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nr 3
283-293
EN
We analyze several "strong meager" properties for filters on the natural numbers between the classical Baire property and a filter being $F_σ$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P⁺-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P⁺-filter. Our motivation lies in understanding the structure of filters generated by complements of members of a maximal almost disjoint family.
3
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Topology from Neighbourhoods

100%
Coghetto R.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
289-296
EN
Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology. We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].
4
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On GE-algebras

100%
Bandaru R., Saeid A. B., Jun Y. B.
Bulletin of the Section of Logic
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2021
|
tom 50
|
nr 1
81-96
EN
Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.
5
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Filters of lattices with respect to a congruence

88%
Sambasiva Rao M., El-Mohsen Badawy A.
Discussiones Mathematicae - General Algebra and Applications
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2014
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tom 34
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nr 2
213-219
EN
Some properties of filters on a lattice L are studied with respect to a congruence on L. The notion of a θ-filter of L is introduced and these filters are then characterized in terms of classes of θ. For distributive L, an isomorphism between the lattice of θ-filters of L and the lattice of filters of $L_{/θ}$ is obtained.
6
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Double Sequences and Iterated Limits in Regular Space

88%
Coghetto R.
Formalized Mathematics
|
2016
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tom 24
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nr 3
173-186
EN
First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence [...] converges in “Pringsheim’s sense” but not in Frechet filter on ℕ × ℕ sense. In the next section, we generalize some definitions: “is convergent in the first coordinate”, “is convergent in the second coordinate”, “the lim in the first coordinate of”, “the lim in the second coordinate of” according to [14], in Hausdorff space. Finally, we generalize two theorems: (3) and (4) from [14] in the case of double sequences and we formalize the “iterated limit” theorem (“Double limit” [7], p. 81, par. 8.5 “Double limite” [6] (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) [17] and the corrections B.10 [18].
7
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Compounding Objects

88%
Šikić Z.
Bulletin of the Section of Logic
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2020
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tom 49
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nr 2
EN
We prove a characterization theorem for filters, proper filters and ultrafilters which is a kind of converse of Łoś's theorem. It is more natural than the usual intuition of these terms as large sets of coordinates, which is actually unconvincing in the case of ultrafilters. As a bonus, we get a very simple proof of Łoś's theorem.
8
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Congruences on semilattices with section antitone involutions

75%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
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2010
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tom 30
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nr 2
207-215
EN
We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.
9
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Hilbert algebras as implicative partial semilattices

75%
Cīrulis J.
Open Mathematics
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2007
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tom 5
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nr 2
264-279
EN
The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.
10
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IF-filters of pseudo-BL-algebras

75%
Wojciechowska-Rysiawa M.
Discussiones Mathematicae - General Algebra and Applications
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2015
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tom 35
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nr 2
177-193
EN
Characterizations of IF-filters of a pseudo-BL-algebra are established. Some related properties are investigated. The notation of prime IF- filters and a characterization of a pseudo-BL-chain are given. Homomorphisms of IF-filters and direct product of IF-filters are studied.
11
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Convergent Filter Bases

75%
Coghetto R.
Formalized Mathematics
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2015
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tom 23
|
nr 3
189-203
EN
We are inspired by the work of Henri Cartan [16], Bourbaki [10] (TG. I Filtres) and Claude Wagschal [34]. We define the base of filter, image filter, convergent filter bases, limit filter and the filter base of tails (fr: filtre des sections).
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