Recent zbMATH articles in MSC 06Bhttps://zbmath.org/atom/cc/06B2021-06-15T18:09:00+00:00WerkzeugObject-oriented interval-set concept lattices.https://zbmath.org/1460.681052021-06-15T18:09:00+00:00"Ma, Jianmin"https://zbmath.org/authors/?q=ai:ma.jianmin"Hu, Lingling"https://zbmath.org/authors/?q=ai:hu.lingling"Qian, Yuhua"https://zbmath.org/authors/?q=ai:qian.yuhuaSummary: Formal concept analysis and rough set are two kinds of efficient mathematical tools for data analysis and knowledge discovery. By combining these two theories, object-oriented and property-oriented concept lattices are proposed. Interval set theory is proposed to describe a partially-known concept by a lower bound and a upper bound. In order to obtain the more accurate extension and intension for a partially-known object-oriented concept, we introduce the theory of interval sets into the object-oriented concept lattice, and propose an object-oriented interval-set concept lattice. Properties of them are investigated. Relationships among interval-set concept lattices, object-oriented interval-set concept lattices and property-oriented interval-set concept lattices are discussed. By discussing the relationships between the object-oriented concept lattice and the object-oriented interval-set concept lattice, an approach to construct object-oriented interval-set concept lattices are established.The Fell compactification of a poset.https://zbmath.org/1460.540062021-06-15T18:09:00+00:00"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Harding, J."https://zbmath.org/authors/?q=ai:harding.johnThis paper deals with the hit-or-miss topology on the closed sets of a topological space, and the associated Fell compactification of a locally compact \(T_0\)-space, which is the closure of its image in its hit-or-miss topology. The aim is to investigate the hit-or-miss topology, and associated Fell compactification, when it is applied to the locally compact \(T_0\)-space formed by a poset \(P\) with its Alexandroff topology.
The authors show that the closed sets of \(P\) with the hit-or-miss topology form the Priestley space of the bounded distributive lattice freely generated by the order dual of \(P\). The Fell compactification \(H(P)\) is shown to be the Priestley space of a sublattice of the upsets of \(P\). The restriction of the hit topology to \(H(P)\) is a stable compactification of \(P\); when \(P\) is a chain, this is the least stable compactification of \(P\).
For the entire collection see [Zbl 1448.62015].
Reviewer: Jorge Picado (Coimbra)Properties not retained by pointed enrichments of finite lattices.https://zbmath.org/1460.080012021-06-15T18:09:00+00:00"Basheyeva, Ainur O."https://zbmath.org/authors/?q=ai:basheyeva.aynur-o"Mustafa, Manat"https://zbmath.org/authors/?q=ai:mustafa.manat"Nurakunov, Anvar M."https://zbmath.org/authors/?q=ai:nurakunov.anvar-mLet \(\langle A;\sigma\rangle \) be an algebra of a finite signature \(\sigma \). An algebra \(\langle A;\sigma\cup C\rangle \) with non-empty finite set of constant symbols \(C\) is called a pointed enrichment of \(\langle A;\sigma\rangle \). In this paper, the authors show by two examples of finite lattices that the properties ``to have (not) a finite basis of quasi-identities'' and ``to generate a standard topological quasivariety'' are not preserved with respect to pointed enrichments of finite algebras.
Reviewer: Alexander Ivanovich Budkin (Barnaul)