Recent zbMATH articles in MSC 06Dhttps://www.zbmath.org/atom/cc/06D2022-05-16T20:40:13.078697ZWerkzeug\( \mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posetshttps://www.zbmath.org/1483.060062022-05-16T20:40:13.078697Z"Zhang, Wenfeng"https://www.zbmath.org/authors/?q=ai:zhang.wenfeng.1|zhang.wenfeng"Xu, Xiaoquan"https://www.zbmath.org/authors/?q=ai:xu.xiaoquan.1|xu.xiaoquanIn domain theory, one fundamental result states that a poset is continuous if and only if it is quasicontinuous and meet-continuous. In this paper, the authors study two kinds of distributivity: \(Z\)-quasidistributivity and \(Z\)-meet-distributivity, which are the generalizations of quasicontinuity and meet-continuity. Here, \(Z\) is a subset system, it becomes meaningful when \(Z\) is replaced by adjectives such as ``directed'', ``chain'', ``finite'', etc. Analogous to the above fundamental result, the authors prove that, under some conditions, a poset is \(Z\)-predistributive iff it is \(Z\)-quasidistributive and \(Z\)-meet-distributive.
In order theory, the Dedekind-MacNeille completion is the most well-known completion, which embeds a poset into a complete lattice. The order-theoretical properties which are invariant under the Dedekind-MacNeille completion are called completion-invariant. In this paper, one main result states that \(Z\)-quasidistributivity is a completion-invariant property whenever \(Z\) is completion-stable.
The way-below relation is a fundamental concept in domain theory. Replacing directed sets by \(Z\)-sets, one has the concept of \(Z\)-below. The last main result of this paper: if the \(Z\)-below relation on the subsets of a poset \(P\) has the interpolation property, then \(P\) is embeddable in a cube.
Reviewer: Zhongxi Zhang (Yantai)Frames of continuous functionshttps://www.zbmath.org/1483.060132022-05-16T20:40:13.078697Z"Lowen, Wendy"https://www.zbmath.org/authors/?q=ai:lowen.wendy"Sioen, Mark"https://www.zbmath.org/authors/?q=ai:sioen.mark"Van Den Haute, Wouter"https://www.zbmath.org/authors/?q=ai:van-den-haute.wouterIn this paper, the authors propose a new approach to representing a topological space via a frame of continuous functions with values in what they call a topological frame. A topological frame is a frame \(\mathbb{F}\) equipped with a topology such that the operations \[\wedge\colon \mathbb{F}\times \mathbb{F} \to \mathbb{F}\colon (a,b)\mapsto a\wedge b\] and \[\sup_{i\in I}\colon \mathbb{F}^I\to \mathbb{F}\colon (a_i)_{i\in I}\mapsto \sup_{i\in I} a_i\] are continuous. The idea extends that of pointfree topology of investigating topological spaces via their open-set lattices (which are frames of continuous functions to the Sierpinski space).
The authors investigate properties of a topological space \(X\) via the frame of continuous functions from \(X\) to a topological frame \(\mathbb{F}\), namely the associated notion of sobriety (\(\mathbb{F}\)-sobriety). One of the interesting results provides conditions on \(\mathbb{F}\) ensuring that a Hausdorff topological space is \(\mathbb{F}\)-sober. These conditions are fulfilled as soon as \(\mathbb{F}\) is a chain with \(0\neq 1\) equipped with the Scott topology. Further, \(\mathbb{F}\)-spectra of \(\mathbb{F}\)-function frames are computed for various spaces \(X\) and frames \(\mathbb{F}\) and a number of spaces that are not \(\mathbb{F}\)-sober are exhibited, showing in particular that the Hausdorff condition in the aforementioned result cannot be relaxed to classical sobriety. A final section discusses the relation between \(\mathbb{F}\)-sobriety and the notion of \(\mathbb{F}\)-fuzzy sobriety as considered in [\textit{D. Zhang} and \textit{Y. Liu}, Fuzzy Sets Syst. 76, No. 2, 259--270 (1995; Zbl 0852.54008)].
The paper ends with a brief outline of some open problems.
Reviewer: Jorge Picado (Coimbra)Weakly spatial localeshttps://www.zbmath.org/1483.060142022-05-16T20:40:13.078697Z"Sun, Xiang Rong"https://www.zbmath.org/authors/?q=ai:sun.xiangrong"He, Wei"https://www.zbmath.org/authors/?q=ai:he.wei.2|he.wei|he.wei.3|he.wei.1(no abstract)\(DR_0\) algebras: a class of regular residuated lattices via De Morgan algebrashttps://www.zbmath.org/1483.060152022-05-16T20:40:13.078697Z"Zhang, Xiao Hong"https://www.zbmath.org/authors/?q=ai:zhang.xiaohong"Wei, Ping"https://www.zbmath.org/authors/?q=ai:wei.ping(no abstract)Some remarks on divisible polyhedral MV-algebrashttps://www.zbmath.org/1483.060162022-05-16T20:40:13.078697Z"Lapenta, Serafina"https://www.zbmath.org/authors/?q=ai:lapenta.serafinaSummary: Building on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.On the Riesz structures of a lattice ordered abelian grouphttps://www.zbmath.org/1483.060232022-05-16T20:40:13.078697Z"Lenzi, Giacomo"https://www.zbmath.org/authors/?q=ai:lenzi.giacomoSummary: A Riesz structure on a lattice ordered abelian group \(G\) is a real vector space structure where the product of a positive element of \(G\) and a positive real is positive. In this paper we show that for every cardinal \(k\) there is a totally ordered abelian group with at least \(k\) Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same \(l\)-group with strong unit. This gives a solution to a problem posed by \textit{P. Conrad} in 1975 [J. Aust. Math. Soc., Ser. A 20, 332--347 (1975; Zbl 0317.06014)]. Finally we apply the main result to MV-algebras and Riesz MV-algebras.On continuous functions on \(LG\)-topologyhttps://www.zbmath.org/1483.540012022-05-16T20:40:13.078697Z"Badie, Mehdi"https://www.zbmath.org/authors/?q=ai:badie.mehdi"Shahidikia, Ali"https://www.zbmath.org/authors/?q=ai:shahidikia.ali"Kasiri, Hossein"https://www.zbmath.org/authors/?q=ai:kasiri.hosseinSummary: In this article, we introduce \(OLG\), \(CLG\) and \(LG\) maps in the context of \(LGT\)-spaces (\(l\)-generalized topological spaces, see [\textit{A. R. Aliabad} and \textit{A. Sheykhmiri}, Bull. Iran. Math. Soc. 41, No. 1, 239--258 (2015; Zbl 1345.06007)]), show that they are generalizations of continuous function on \(LGT\)-spaces and some properties of them studied. Also, some generalized notions related to continuous functions as weak topology induced, quotient topology and decomposition topology are introduced and studied and is shown that each decomposition space is an \(LG\)-quotient space.A study of algebras and logics of rough sets based on classical and generalized approximation spaceshttps://www.zbmath.org/1483.684052022-05-16T20:40:13.078697Z"Kumar, Arun"https://www.zbmath.org/authors/?q=ai:kumar.arun-m|kumar.arun-nSummary: The seminal work of
\textit{Z. Pawlak} [Int. J. Comput. Inform. Sci. 11, 341--356 (1982; Zbl 0501.68053)]
on rough set theory has attracted the attention of researchers from various disciplines. Algebraists introduced some new algebraic structures and represented some old existing algebraic structures in terms of algebras formed by rough sets. In Logic, the rough set theory serves the models of several logics. This paper is an amalgamation of algebras and logics of rough set theory. We prove a structural theorem for Kleene algebras, showing that an element of a Kleene algebra can be looked upon as a rough set in some appropriate approximation space. The proposed propositional logic \(\mathcal{L}_K\) of Kleene algebras is sound and complete with respect to a 3-valued and a rough set semantics.
This article also investigates some negation operators in classical rough set theory, using Dunn's approach. We investigate the semantics of the Stone negation in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property in \(K_-\) frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras, and regular double Stone algebras. As the perp semantics provides a Kripke type semantics for logics with negations, exploiting this feature, we obtain duality results for several classes of algebras and corresponding frames.
In another part of this article, we propose a granule-based generalization of rough set theory. We obtain representations of distributive lattices (with operators) and Heyting algebras (with operators). Moreover, various negations appear from this generalized rough set theory and achieved new positions in Dunn's Kite of negations.
For the entire collection see [Zbl 1475.68025].An abstract theory of physical measurementshttps://www.zbmath.org/1483.810152022-05-16T20:40:13.078697Z"Resende, Pedro"https://www.zbmath.org/authors/?q=ai:resende.pedroSummary: The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a fundamental process via which a finite amount of classical information is produced. This translates into an algebraic and topological definition of \textit{measurement space} that caters for the distinction between quantum and classical measurements and allows a notion of observer to be derived.