Recent zbMATH articles in MSC 06https://www.zbmath.org/atom/cc/062021-03-30T15:24:00+00:00WerkzeugSums of strongly \(z\)-ideals and prime ideals in \(\mathcal{RL}\).https://www.zbmath.org/1455.060072021-03-30T15:24:00+00:00"Estaji, A. A."https://www.zbmath.org/authors/?q=ai:estaji.ali-akbar"Karimi Feizabadi, A."https://www.zbmath.org/authors/?q=ai:karimi-feizabadi.abolghasem"Robat Sarpoushi, M."https://www.zbmath.org/authors/?q=ai:robat-sarpoushi.maryamSummary: It is well-known that the sum of two \(z\)-ideals in \(C(X)\) is either \(C(X)\) or a \(z\)-ideal. The main aim of this paper is to study the sum of strongly \(z\)-ideals in \(\mathcal{RL}\), the ring of real-valued continuous functions on a frame \(L\). For every ideal \(I\) in \(\mathcal{RL}\), we introduce the biggest strongly \(z\)-ideal included in \(I\) and the smallest strongly \(z\)-ideal containing \(I\), denoted by \(I^{sz}\) and \(I_{sz}\), respectively. We study some properties of \(I^{sz}\) and \(I_{sz}\). Also, it is observed that the sum of any family of minimal prime ideals in the ring \(\mathcal{RL}\) is either \(\mathcal{RL}\) or a prime strongly \(z\)-ideal in \(\mathcal{RL}\). In particular, we show that the sum of two prime ideals in \(\mathcal{RL}\) such that are not a chain, is a prime strongly \(z\)-ideal.A compact representation for minimizers of \(k\)-submodular functions (extended abstract).https://www.zbmath.org/1455.901342021-03-30T15:24:00+00:00"Hirai, Hiroshi"https://www.zbmath.org/authors/?q=ai:hirai.hiroshi"Oki, Taihei"https://www.zbmath.org/authors/?q=ai:oki.taiheiIn the paper the authors compactly represent the minimizers of a \(k\)-submodular function by posets with inconsistent pairs and completely characterize the corresponding class of posets. They propose a fast algorithm to obtain a representation of the minimizer set for the cases when either a minimizing oracle can be used, or a \(k\)-submodular function is represented by a network, or a \(k\)-submodular function is the objective function of the relaxed multiway cut problem. An efficient algorithm to enumerate all maximal minimizers is developed. The possible applications of the proposed enumeration algorithm to the computer vision problems are discussed.
For the entire collection see [Zbl 1346.90004].
Reviewer: Svetlana A. Kravchenko (Minsk)On the properties of zero-divisor graphs of posets.https://www.zbmath.org/1455.050172021-03-30T15:24:00+00:00"Afkhami, Mojgan"https://www.zbmath.org/authors/?q=ai:afkhami.mojgan"Khashyarmanesh, Kazem"https://www.zbmath.org/authors/?q=ai:khashyarmanesh.kazem"Shahsavar, Faeze"https://www.zbmath.org/authors/?q=ai:shahsavar.faezeSummary: We determine the cut vertices in the zero-divisor graphs of posets and study the posets with end-regular zero-divisor graph. Also, we investigate the zero-divisor graph of the product of two posets. In particular, we determine all posets with planar and outerplanar zero-divisor graphs.Lattices of subspaces of vector spaces with orthogonality.https://www.zbmath.org/1455.060062021-03-30T15:24:00+00:00"Chajda, Ivan"https://www.zbmath.org/authors/?q=ai:chajda.ivan"Länger, Helmut"https://www.zbmath.org/authors/?q=ai:langer.helmut-mConsider a vector space over a field and the set of all subspaces, partially ordered by set theoretic inclusion. Then it is a complete modular lattice.
The authors investigate under which conditions this lattice is orthocomplemented with respect to the orthogonality operation.
After defining closed subspaces using this orthogonality operation, the authors introduce splitting subspaces as closed subspaces and they prove that the poset of splitting subspaces and the poset of projections are isomorphic orthomodular posets.
Reviewer: Giuseppina Barbieri (Fisciano)Cayley posets.https://www.zbmath.org/1455.060022021-03-30T15:24:00+00:00"García-Marco, Ignacio"https://www.zbmath.org/authors/?q=ai:garcia-marco.ignacio"Knauer, Kolja"https://www.zbmath.org/authors/?q=ai:knauer.kolja-b"Mercui-Voyant, Guillaume"https://www.zbmath.org/authors/?q=ai:mercui-voyant.guillaumeLet \((X, S)\) be a semigroup (right) act, i.e., \(X\)
is a set and \((S,\cdot)\) a semigroup with an operation, such that \(xs \in X\) and
\((xs)s'=x(s \cdot s')\) for all \(x \in X\) and \(s, s'\in S\). The binary relation
\(\leq_S\) on \(X\) is defined by \(x \leq_S y\) if and only if there is an \(s \in S\) such that \(xs = y\).
The paper under review studies acts, such that \(\leq _S\) is a partial order relation. Such objects arise also naturally from considering
(relative) Green's relations on semigroups. An important class of Cayley
posets are numerical semigroups, and more generally
numerical semigroups with torsion, and affine semigroups.
Reviewer: László A. Székely (Columbia)Algorithm for generating finite totally ordered monoids.https://www.zbmath.org/1455.060082021-03-30T15:24:00+00:00"Petrík, Milan"https://www.zbmath.org/authors/?q=ai:petrik.milan"Vetterlein, Thomas"https://www.zbmath.org/authors/?q=ai:vetterlein.thomasSummary: The semantics of fuzzy logic is typically based on negative totally ordered monoids. This contribution describes an algorithm generating in a step-wise fashion all finite structures of this kind.
For the entire collection see [Zbl 1385.68004].Chapoton triangles for nonkissing complexes.https://www.zbmath.org/1455.050802021-03-30T15:24:00+00:00"Garver, Alexander"https://www.zbmath.org/authors/?q=ai:garver.alexander"McConville, Thomas"https://www.zbmath.org/authors/?q=ai:mcconville.thomasSummary: We continue the study of the nonkissing complex that was introduced by \textit{T. K. Petersen} et al. [J. Algebra 324, No. 5, 951--969 (2010; Zbl 1203.13026)] and was studied lattice-theoretically by the second author. We introduce a theory of Grid-Catalan combinatorics, given the initial data of a nonkissing complex, and show how this theory parallels the well-known Coxeter-Catalan combinatorics. In particular, we present analogues of Chapoton's \(F\)-triangle, \(H\)-triangle, and \(M\)-triangle and give both combinatorial and lattice-theoretic interpretations of the objects defining these polynomials. In our Grid-Catalan setting, we prove that our analogue of Chapoton's \(F\)-triangle and \(H\)-triangle identity holds, and we conjecture that our analogue of Chapoton's \(F\)-triangle and \(M\)-triangle identity also holds.Łukasiewicz public announcement logic.https://www.zbmath.org/1455.030172021-03-30T15:24:00+00:00"Cabrer, Leonardo"https://www.zbmath.org/authors/?q=ai:cabrer.leonardo-manuel"Rivieccio, Umberto"https://www.zbmath.org/authors/?q=ai:rivieccio.umberto"Rodriguez, Ricardo Oscar"https://www.zbmath.org/authors/?q=ai:rodriguez.ricardo-oscarSummary: In this work we lay a theoretical framework for developing dynamic epistemic logics in a many-valued setting. We consider in particular the logic of Public Announcements, which is one of the simplest and best-known dynamic epistemic systems in the literature. We show how to develop a Public Announcement Logic based on finite-valued Łukasiewicz modal logic. We define our logic through a relational semantics based on many-valued Kripke models, and also introduce an alternative but equivalent algebra-based semantics using MV-algebras endowed with modal operators. We provide a Hilbert-style calculus for our logic and prove completeness with respect to both semantics.
For the entire collection see [Zbl 1385.68004].Towards Galois connections over positive semifields.https://www.zbmath.org/1455.682032021-03-30T15:24:00+00:00"Valverde-Albacete, Francisco J."https://www.zbmath.org/authors/?q=ai:valverde-albacete.francisco-jose"Peláez-Moreno, Carmen"https://www.zbmath.org/authors/?q=ai:pelaez-moreno.carmenSummary: In this paper we try to extend the Galois connection construction of \(\mathcal K\)-Formal Concept Analysis to handle semifields which are not idempotent. Important examples of such algebras are the extended non-negative reals and the extended non-negative rationals, but we provide a construction that suggests that such semifields are much more abundant than suspected. This would broaden enormously the scope and applications of \(\mathcal K\)-Formal Concept Analysis.
For the entire collection see [Zbl 1385.68004].Reduct-irreducible \(\alpha\)-cut concept lattices: an efficient reduction procedure to multi-adjoint concept lattices.https://www.zbmath.org/1455.682002021-03-30T15:24:00+00:00"Cornejo, M. Eugenia"https://www.zbmath.org/authors/?q=ai:cornejo.maria-eugenia"Medina, Jesús"https://www.zbmath.org/authors/?q=ai:medina.jesus"Ramírez-Poussa, Eloísa"https://www.zbmath.org/authors/?q=ai:ramirez-poussa.eloisaSummary: The computation of fuzzy concept lattices is really complex. Hence, looking for mechanisms in order to reduce this complexity is fundamental. This paper presents a new efficient mechanism which combines two procedures. First of all, an attribute reduction is given, which removes the unnecessary attributes, and then a reduction based on a truth degree is applied, which removes the fuzzy attributes with low weight. Different interesting properties and examples of this mechanism are also introduced.
For the entire collection see [Zbl 1385.68004].On semilattice of Linux processes' states.https://www.zbmath.org/1455.680412021-03-30T15:24:00+00:00"Efanov, Nikolaĭ Nikolaevich"https://www.zbmath.org/authors/?q=ai:efanov.nikolai-nikolaevichSummary: The paper discusses a set of states of Linux processes as data structure, which is used in the task of process-tree reconstruction in Unix-like operating systems. The purpose of the study is to analyze dependencies in such structure, to introduce the natural order of dependencies, to propose the class of such reconstruction structure as upper complete semilattice. Following from the technical properties of the applied problem attributes' hierarchy allow to introduce additional restrictions on the minimum upper bounds in such semilattice.
Constraints are formally described as suitable pre-closure and closure operators. The constraints implies the necessary condition for the correctness of the process tree. Based on the properties of points returned by the proposed operators and system call execution scheme, a sufficient condition for correctness is given. The introduced conditions form the criterion for process-tree correctness, which can be useful in such tasks as generating tests for checkpoint-restore in Unix-like operating systems, anomalies detection, increasing portability and reliability of software. Dependency schemes between attributes that impose particular constraints on the reconstructing set are also shown. Opened questions are also highlighted and further steps are suggested.On generalized fuzzy sets in ordered LA-semihypergroups.https://www.zbmath.org/1455.060112021-03-30T15:24:00+00:00"Gulistan, Muhammad"https://www.zbmath.org/authors/?q=ai:gulistan.muhammad"Yaqoob, Naveed"https://www.zbmath.org/authors/?q=ai:yaqoob.naveed"Kadry, Seifedine"https://www.zbmath.org/authors/?q=ai:kadry.seifedine"Azhar, Muhammad"https://www.zbmath.org/authors/?q=ai:azhar.muhammad-razwan|azhar.muhammad-naeemThe authors introduce the notions of generalized fuzzy hyperideals, generalized fuzzy bi-hyperideals, and generalized fuzzy normal bi-hyperideals in an ordered nonassociative and non-commutative algebraic structure, namely an ordered LA-semihypergroup using the notion of generalized fuzzy sets and characterize these hyperideals. Also, they show that the set of all fuzzy hyperideals becomes an ordered LA-semihypergroup.
Reviewer: Dariush Heidari (Mahallat)The diagonal of the associahedra.https://www.zbmath.org/1455.180142021-03-30T15:24:00+00:00"Masuda, Naruki"https://www.zbmath.org/authors/?q=ai:masuda.naruki"Thomas, Hugh"https://www.zbmath.org/authors/?q=ai:thomas.hugh-ross"Tonks, Andy"https://www.zbmath.org/authors/?q=ai:tonks.andy"Vallette, Bruno"https://www.zbmath.org/authors/?q=ai:vallette.brunoThis paper has a threefold purpose.
\begin{itemize}
\item[1.] to introduce a general machinery to solve the problem of the approximation of the diagonal of \textit{face-coherent families of polytopes} (\S 2),
\item[2.] to give a complete proof for the case of the \textit{associahedra} (Theorem 1), and
\item[3.] to popularize the resulting \textit{magical formula} (Theorem 2).
\end{itemize}
The problem of the approximation of the diagonal of the associahedra lies at the crossroads of three clusters of domains.
\begin{itemize}
\item[1.] There are mathematicians inclined to apply it in their work of computing the homology of fibered spaces in algebraic topology [\textit{E. H. Brown jun.}, Ann. Math. (2) 69, 223--246 (1959; Zbl 0199.58201); \textit{A. Prouté}, Repr. Theory Appl. Categ. 2011, No. 21, 99 p. (2011; Zbl 1245.55007)], to construct tensor products of string field theories [\textit{M. R. Gaberdiel} and \textit{B. Zwiebach}, Nucl. Phys., B 505, No. 3, 569--624 (1997; Zbl 0911.53044); Phys. Lett., B 410, No. 2--4, 151--159 (1997; Zbl 0911.53046)], or to consider the product of Fukaya \(\mathcal{A}_{\infty}\)-categories in symplectic geometry [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{L. Amorim}, Int. J. Math. 28, No. 4, Article ID 1750026, 38 p. (2017; Zbl 1368.53057)].
\item[2.] The analogous result is known, within the kingdom of operad theory and homotopical algebra, in the differential graded context [\textit{S. Saneblidze} and \textit{R. Umble}, Homology Homotopy Appl. 6, No. 1, 363--411 (2004; Zbl 1069.55015); \textit{L. J. Billera} and \textit{B. Sturmfels}, Ann. Math. (2) 135, No. 3, 527--549 (1992; Zbl 0762.52003)].
\item[3.] This result is appreciated conceptualy as a new development in the theory of \textit{fiber polytopes} [loc. cit.] by combinatorists and
discrete geometers.
\end{itemize}
The possible ways of iteraring a binary product are to be encoded by planar binary trees, the associativity relation being interpreted as an order relation, which encouraged Dov Tamari to introduce the so-called \textit{Tamari lattice} [\textit{D. Tamari}, Nieuw Arch. Wiskd., III. Ser. 10, 131--146 (1962; Zbl 0109.24502)]. These lattices are to be realized by associahedra in the sense that their \(1\)-skeleton is the Hasse diagram of the Tamari lattice [\textit{C. Ceballos} et al., Combinatorica 35, No. 5, 513--551 (2015; Zbl 1389.52013)]. For loop spaces, composition fails to be strictly associative due to the different parametrizations, but this failure is governed by an infinite sequence of higher homotopies, which was made precise by \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)], introducing a family of curvilinear polytopes called the \textit{Stascheff polytopes}, whose combinatorics coincides with the associahedra. Stascheff's work opened the door to the study of homotopical algebra by means of operad-like objects, summoning [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0285.55012)] and [\textit{J. P. May}, The geometry of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009)] in particular, the latter of which introduced the \textit{little disks} operads playing a key role in many domains nowadays. In dimension \(1\), this gives the \textit{little intervals} operad, a finite-dimensional topological operad pervious to Stascheff's theory, whose operad structure is given by scaling a configuration of intervals in order to insert it into another interval. So far there has been no operad structure on any family of convex polytopal realizations of the associahedra in the literature, though there should be a rainbow bridge
between operad theory as well as homotopy theories on the one hand and combinatorics as well as discrete geometry on the other.
Since the set-theoretic diagonal of a polytope fails to be cellular in general, the authors need to find a \textit{cellular approximation to the
diagonal}, that is to say, a cellular map from the polytope to its cartesian square homotopic to the diagonal. For a coherent family of polytopes, it is highly challenging to find a family of diagonals compatible with the combinatorics of faces. In the case of the first face-coherent family of polytopes, the geometric simplices, such a diagronal map is given by the classical \textit{Alexander-Whitney map} [\textit{S. Eilenberg} and \textit{J. A. Zilber}, Am. J. Math. 75, 200--204 (1953; Zbl 0050.17301); \textit{N. E. Steenrod}, Ann. Math. (2) 48, 290--320 (1947; Zbl 0030.41602)]. For the next family given by cubes, a coassociative approximation to the diagonal is straightforward [\textit{J.-P. Serre}, Ann. Math. (2) 54, 425--505 (1951; Zbl 0045.26003)]. The associahedra form the face-coherent family of polytopes coming next in terms of further truncations of the simplices or of combinatorial complexity. While a face of a simplex or a cube is a simplex or a cube of lower dimension, a face of an associahedra is a product of associahedra of lower dimensions, which makes the problem of the
approximation of the diagonal in this turn highly intricate. The two-fold principal result of the paper (Theorem 1) is an explicit operad structure on the \textit{Loday realizations} of the associahedra together with a compatible approximation to the diagonal.
There is a dichotomy between pointwise and cellular formulas. To investigate
their relationship and to make precise the various face-coherent properties,
the authors introduce the \textit{category of polytopes with subdivision}. The
definition of the diagonal maps comes from the theory of fiber polytopes so
that an induced polytopal subdivision of the associahedra is obtained, for
which the authors establishes a magical formula in the verbalism of Jean-Louis
Loday. It is made up of the pairs of cells of matching dimensions and
comparable order under the Tamari order (Theorem 2).
A synopsis of the paper consisting of four sections goes as follows. \S 1
recalls the main relevant notions, introducing the category of polytopes in
which the authors work. \S 2 gives a canonical definition of the diagonal map
for positively oriented polytopes, addressing their cellular properties. \S 3
endows the family of Loday realizations of the associahedra with a
nonsymmetric operad structure compatible with the diagonal maps. \S 4
establishes the magical celluar formula for the diagonal map of the associahedra.
Reviewer: Hirokazu Nishimura (Tsukuba)Spectral spaces.https://www.zbmath.org/1455.540012021-03-30T15:24:00+00:00"Dickmann, Max"https://www.zbmath.org/authors/?q=ai:dickmann.max-a"Schwartz, Niels"https://www.zbmath.org/authors/?q=ai:schwartz.niels"Tressl, Marcus"https://www.zbmath.org/authors/?q=ai:tressl.marcusAlthough the theory of spectral spaces has been an active topic
of research for more than eighty years and appeared in more than 1000 research articles, this is the first monograph devoted to the topic. The authors ambitiously pursue a variety of goals in the book, with a foundational goal of giving a coherent and reasonably comprehensive treatment of the
\emph{topological} theory of spectral spaces. Additional important goals include accessibility of the material, particularly including introductory material
at a level accessible to graduate students and material of potential interest to other researchers with overlapping interests in some of the various specialized topics treated. Accessibility is also enhanced by inclusion of a wide variety of examples and concrete descriptions of various constructions. As the overall
theory is scattered through a wide variety of scholarly writings, the authors have also sought to find connections, fill in gaps, and provide a more comprehensive approach, thus enriching, not just recording earlier work. But beyond all this they present material resulting from their unified approach that belongs to the frontiers of current
research. Thus in the end the book has more the flavor of a research monograph
and reference source. The latter is important, given the growth of the field and the previous lack of such a source.
Although spectral spaces form a special class of topological spaces, their historical
roots lie in algebraic settings. Indeed their name ``spectral spaces'' derives from
the fact that they are precisely the spaces that arise as Zariski spectra of commutative rings, first
famously shown by \textit{M. Hochster} [Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)]. They first appeared in the work of \textit{M. H. Stone} [Čas. Mat. Fys. 67, 1--25 (1937; JFM 63.0830.01)], in which he showed the duality of bounded distributive lattices
and spectral spaces, a follow-up to his famous earlier work on what we know as
the Stone duality of Boolean lattices. The whole theory received a major boost in the
1960s through Alexander Grothendieck's introduction in algebraic geometry of affine schemes,
which, loosely speaking, allowed a commutative ring to be viewed as a ring of functions on its
Zariski spectrum, an idea that has had far reaching consequences.
A spectral space is a compact sober space with a basis of compact, open subsets closed under
finite intersection. (Here the reviewer replaces the authors' ``quasi-compact'' with the more common
``compact.'') The topology of spectral spaces reflects the fact that they are \(T_0\), typically not Hausdorff,
and hence one has important features such as the order of specialization, a naturally associated partial order.
With respect to this order one has topologies with the reversed order of specialization and ``patch'' topologies generated
by a topology and one with reversed order. An important one with reversed order is the inverse topology, obtained by
taking the open compact subsets as the closed sets. The join or patch of these two topologies together with the partial order of specialization
yields with what is called a Priestley space, a special compact Hausdorff partially ordered space.
It turns out that spectral spaces and Priestley spaces are the same structures viewed from different perspectives, and both are important.
Thus in some sense spectral spaces live in both the \(T_0\)-world and the Hausdorff world. Finally the spectral spaces form the
objects of a category, with the morphisms all continuous maps between objects with the property that the inverse
image of a compact open set is again compact. All this and more are treated in the first introductory chapter.
The book breaks down roughly into two parts with the first part, Chapters 1 through 6, devoted to a wide variety of
topological properties and constructions in the category of spectral spaces. (Chapter 3 is something of an exception as it reviews and elaborates
on the Stone duality of bounded distributive lattices with dual category the category of spectral spaces.)
The second part, consisting of the remaining chapters, looks at various other settings in which spectral spaces appear, and treats
the theory in those various contexts and settings.
Chapter 7 treats intrinsic topologies on a partially ordered set, topologies defined directly from the partial order.
An important one, especially in connection with spectral spaces, is the Scott topology in which the closed sets of a partially ordered set
are the lower or down sets that are also closed under taking directed suprema. The authors identify the conditions for
the Scott topology to be spectral and include other considerations involving intrinsic topologies and spectral spaces. Chapter
10 treats the construction of infinite colimits in the category of spectral spaces.
From Chapter 8 through 13, with the exception of Chapter 10, various special classes of spectral spaces are considered, and these
chapters will be of interest according to the readers' background and research interests.
These special classes are frequently coming from various other mathematical settings and represent the diverse appearance
of frames. Chapter 8 considers special subclasses of spectral spaces such as Noetherian spaces arising as spectra of Noetherian rings and
Heyting spaces arising as spectra of Heyting algebras. Chapter 9 treats localic spaces, spaces arising as spectra of locales. Chapter 11 considers
spectral reflections of topological spaces and other relations between the category of spectral spaces and other topological categories and categories
of partially ordered sets. Chapter 12 develops the theory of the Zariski spectrum of a ring, which as mentioned previously is always a spectral space,
and Chapter 13 turns to the real spectrum and connections with real algebraic geometry. Chapter 14 closes the book with appearance of spectral
spaces in Model Theory.
This quick overview should convince the reader that this book is a valuable resource for anyone seriously interested in the theory of spectral spaces
and represents a substantial addition to the literature on the subject.
Reviewer: Jimmie Lawson (Baton Rouge)Organized collapse. An introduction to discrete Morse theory.https://www.zbmath.org/1455.570012021-03-30T15:24:00+00:00"Kozlov, Dmitry N."https://www.zbmath.org/authors/?q=ai:feichtner-kozlov.dmitryThis book provides an overview of discrete Morse theory, in the style of Forman. Discrete Morse theory is a powerful tool that has emerged in the last few decades, as a discretized version of classical smooth Morse theory. As the title of the book (``Organized Collapse'') suggests, the main idea behind discrete Morse theory involves ``collapsing'' parts of a complex in an ``organized'' way, to get a smaller complex that is easier to understand, while retaining many features of the original. Discrete Morse theory has proven to be a crucial tool in applied topology, and this book serves as an excellent reference for readers at any level of discrete Morse theoretic understanding.
There are four parts to the book. The first two parts focus on algebraic topology and specifically homology theory, and are very detailed and thorough. The author points out that anyone not interested in the nitty-gritty behind-the-scenes material of why discrete Morse theory works are free to skip the more difficult aspects of these parts. The core of the book is the third part, on basic discrete Morse theory. This is where the theory is set up, the basic properties are presented and proved, and the most standard advanced topics are covered, e.g., the so called Morse complex, discrete vector fields, and shellability. Finally, the fourth part focuses on more advanced generalizations of discrete Morse theory. This includes an algebraic form applied directly to chain complexes with no reference to an underlying simplicial complex, a combinatorial version that deals with posets, and a version that works on (not necessarily simplicial) regular CW-complexes. The final chapter discusses connections to persistent homology, one of the most prominent modern topics in applied topology.
It is easy to picture using this book as the basis of a graduate topics course. The heavy focus on algebraic topology in the first two parts also makes an argument for its potential use as a reference for a first course on homology theory. All in all it is an excellent addition to the literature on discrete Morse theory.
Reviewer: Matthew Zaremsky (Albany)A completion for distributive nearlattices.https://www.zbmath.org/1455.060042021-03-30T15:24:00+00:00"González, Luciano J."https://www.zbmath.org/authors/?q=ai:gonzalez.luciano-javier"Calomino, Ismael"https://www.zbmath.org/authors/?q=ai:calomino.ismaelBy a polarity a triple \((X,Y,R)\) is meant where \(X,Y\) are nonempty sets and \(R\) is a binary relation between \(X\) and \(Y\). For a poset \(P\), a completion of \(P\) is a pair \((L,e)\) where \(L\) is a complete lattice and \(e\) is an order embedding of \(P\) into \(L\). A collection \(F\) of upsets of \(P\) is \(standard\) if it contains all principal filters of \(P\), dually for a collection \(I\) of downsets of \(P\). A polarity is \(standard\) if it is of the form \((F,I,R)\) for standard collections \(F\) and \(I\). The authors introduce the so-called \((F,I)\)-compact and \((F,I)\)-dense polarities and the so-called \((F,I)\)-completion They prove that every distributive nearlattice can be embedded into a complete distributive lattice via an \((F,I)\)-completion and presented a connection with free distributive lattice extension. They study how an \(n\)-ary operation can be extended on a distributive nearlattice.
Reviewer: Ivan Chajda (Přerov)General non-commutative locally compact locally Hausdorff Stone duality.https://www.zbmath.org/1455.060052021-03-30T15:24:00+00:00"Bice, Tristan"https://www.zbmath.org/authors/?q=ai:bice.tristan-matthew"Starling, Charles"https://www.zbmath.org/authors/?q=ai:starling.charlesClassical Stone duality constructs a contravariant equivalence between the categories of \(0\)-dimensional compact Hausdorff
spaces (plus continuous maps) and Boolean algebras (plus homomorphisms). On the one side there is a category of topological structures and continuous homomorphisms; on the other there is a category whose objects constitute an elementary productive class -- in the model-theoretic sense -- of finitary relational structures. (Another famous example of this phenomenon is Pontryagin duality between the compact Hausdorff and the discrete abelian groups.) In the early 1980s, this reviewer asked whether Stone duality could be properly extended to a full subcategory of the compact Hausdorff spaces -- in such a way that the dual category is still an elementary productive class -- and \textit{B. Banaschewski} [Can. J. Math. 36, 1113--1118 (1984; Zbl 0561.18004)] gave an elegant proof that it could not. In the present paper, the authors take a different approach to the problem of going beyond \(0\)-dimensionality and set about showing that ``certain bases of general locally compact locally Hausdorff étale groupoids are dual to a natural first order finitely axiomatizable class of inverse semigroups.'' The details are quite involved, and the interested reader is invited to consult the paper.
Reviewer: Paul Bankston (Milwaukee)On a preorder relation for Schur-convex functions and a majorization inequality for their gradients and divergences.https://www.zbmath.org/1455.260102021-03-30T15:24:00+00:00"Niezgoda, Marek"https://www.zbmath.org/authors/?q=ai:niezgoda.marekAfter proposing a preordering for Schur-convex functions on \(\mathbb{R}^n\), a majorization statement involving gradients and divergences of two Gâteaux differentiable Schur-convex functions whose difference is Schur-convex as well is provided. Various implications of this result are provided, in particular for \(c\)-strongly convex functions (for some positive real \(c\)).
Reviewer: Sorin-Mihai Grad (Wien)Fixed point property for finite ordered sets that contain no crowns with 6 or more elements.https://www.zbmath.org/1455.060012021-03-30T15:24:00+00:00"Schröder, Bernd S. W."https://www.zbmath.org/authors/?q=ai:schroder.bernd-s-w.1|schroder.bernd-s-wA poset has the fixed point property if every endomorphism
has a fixed point. The problem of determining whether a
finite poset has the fixed point property is co-NP-complete.
This paper proves that the problem of determining whether
a finite poset which omits crowns of six or more elements
has the fixed point property is in P.
This result is established by first proving
that every finite, connected poset which omits
crowns of six or more elements either has
(i) an element of rank one that has a
unique lower cover or (ii) a retractable minimal element.
Reviewer: Keith Kearnes (Boulder)Some properties of the weak product of graphs on lattices.https://www.zbmath.org/1455.050652021-03-30T15:24:00+00:00"Nimbhorkar, S. K."https://www.zbmath.org/authors/?q=ai:nimbhorkar.shriram-khanderao"Borsarkar, U. R."https://www.zbmath.org/authors/?q=ai:borsarkar.uttara-rIn this paper, the authors studied the atom-based graphs associated with lattices. Let \(L\) be a lattice. Associate a simple graph with \(L\), whose vertex set is the set of all nonzero elements in \(L\) and \(a, b \in L\) are adjacent if and
only if \(a \wedge b\) is an atom in \(L\). This graph is known as the atom-based graph of \(L\). The authors studied the weak product of the atom-based graphs, the incomparability graphs and the zero-divisor graphs of lattices. Also, the number of edges in the weak product of such graphs is found.
Reviewer: Vinayak Joshi (Pune)One hundred twenty-seven subsemilattices and planarity.https://www.zbmath.org/1455.060032021-03-30T15:24:00+00:00"Czédli, Gábor"https://www.zbmath.org/authors/?q=ai:czedli.gaborA semilattice is called planar if it has a Hasse diagram that is a planar representation of a graph. Every semilattice with at most 7 elements is planar. Every 8-element semilattice with at least 121 subsemilattices is planar. The author proves that finite semilattices with many subsemilattices are planar. The main resultis: Theorem. Let \(L\) be a semilattice having \(n\) elements. If \(L\) has at least \(127\cdot 2^{n-8}\) subsemilattices then it is planar. Moreover, for \(n\) greater than 8 this result is sharp because there is a non-planar semilattice with exactly \(127\cdot 2^{n-8} - 1\) subsemilattices.
Reviewer: Ivan Chajda (Přerov)Free sequences in \({\mathscr{P}}( \omega) /\mathrm{fin}\).https://www.zbmath.org/1455.030622021-03-30T15:24:00+00:00"Chodounský, David"https://www.zbmath.org/authors/?q=ai:chodounsky.david"Fischer, Vera"https://www.zbmath.org/authors/?q=ai:fischer.vera"Grebík, Jan"https://www.zbmath.org/authors/?q=ai:grebik.janA free sequence in a Boolean algebra, as defined in [\textit{J. D. Monk}, Commentat. Math. Univ. Carol. 52, No. 4, 593--610 (2011; Zbl 1249.06034)],
is a sequence \(\langle a_\alpha:\alpha<\gamma\rangle\) of elements such that
for every \(\beta\le\gamma\) the set
\(\{a_\alpha:\alpha<\beta\}\cup\{a_\alpha':\beta\le\alpha<\gamma\}\)
is centered.
It is maximal if it has no free end-extension.
Though the order(-type) of the sequence is important in this definition
the cardinal number \(\mathfrak f(B)\) is defined to be the
minimum \textit{cardinality\/} of a free sequence in the algebra \(B\);
if \(B=\mathcal{P}(\omega)/\mathrm{fin}\), then one simply writes \(\mathfrak f\).
After making some remarks on free sequences in \(\mathcal{P}(\omega)/\mathrm{fin}\)
and indicating how much is still unknown the authors show the consistency
of \(\mathfrak i=\mathfrak f<\mathfrak u\), where \(\mathfrak i\) is the minimum cardinality
of a maximal independent family and \(\mathfrak u\) is the minimum character
of an ultrafilter.
They show that this holds in \textit{S. Shelah}'s model for \(\mathfrak i<\mathfrak u\)
from [Arch. Math. Logic 31, No. 6, 433--443 (1992; Zbl 0785.03029)] and give a self-contained presentation of this model.
An independent family is a free sequence no matter how it is ordered and
one would expect some relation between \(\mathfrak i\) and \(\mathfrak f\) to hold;
the Miller model satisfies \(\mathfrak f<\mathfrak i\) but it is not (yet) known
whether \(\mathfrak i<\mathfrak f\) is consistent.
Reviewer: K. P. Hart (Delft)Int-soft implicative hyper BCK-ideals in hyper BCK-algebras.https://www.zbmath.org/1455.060102021-03-30T15:24:00+00:00"Borzooei, Rajab Ali"https://www.zbmath.org/authors/?q=ai:borzooei.rajab-ali"Xin, Xiao Long"https://www.zbmath.org/authors/?q=ai:xin.xiaolong"Roh, Eun Hwan"https://www.zbmath.org/authors/?q=ai:roh.eun-hwan"Jun, Young Bae"https://www.zbmath.org/authors/?q=ai:jun.young-baeRelations between various types of int-soft hyper BCK-ideals are discussed.
Conditions for an int-soft hyper BCK-ideal to be an int-soft weak
implicative hyper BCK-ideal are provided. Using an int-soft weak
implicative hyper BCK-ideal, a new int-soft weak implicative hyper
BCK-ideal is established.
Reviewer: Wiesław A. Dudek (Wrocław)On perfect poset codes.https://www.zbmath.org/1455.942232021-03-30T15:24:00+00:00"Panek, Luciano"https://www.zbmath.org/authors/?q=ai:panek.luciano"Pinheiro, Jerry Anderson"https://www.zbmath.org/authors/?q=ai:pinheiro.jerry-anderson"Alves, Marcelo Muniz"https://www.zbmath.org/authors/?q=ai:alves.marcelo-muniz-silva"Firer, Marcelo"https://www.zbmath.org/authors/?q=ai:firer.marceloSummary: We consider on \( \mathbb{F}_q^n \) metrics determined by posets and classify the parameters of 1-perfect poset codes in such metrics. We show that a code with same parameters of a 1-perfect poset code is not necessarily perfect, however, we give necessary and sufficient conditions for this to be true. Furthermore, we characterize the unique way up to a labeling on the poset, considering some conditions, to extend an \( r \)-perfect poset code over \( \mathbb{F}_q^n \) to an \( r \)-perfect poset code over \( \mathbb{F}_q^{n+m} \).A method for bi-decomposition of partial Boolean functions.https://www.zbmath.org/1455.942332021-03-30T15:24:00+00:00"Pottosin, Yu. V."https://www.zbmath.org/authors/?q=ai:pottosin.yu-vSummary: A method for bi-decomposition of incompletely specified (partial) Boolean functions is suggested. The problem of bi-decomposition is reduced to the problem of two-block weighted covering a set of edges of a graph of rows orthogonality of a ternary or binary matrix that specify a given function, by complete bipartite subgraphs (bicliques). Each biclique is assigned in a certain way with a set of arguments of the given function, and the weight of a biclique is the cardinality of this set. According to each of bicliques, a Boolean function is constructed whose arguments are the variables from the set, which is assigned to the biclique. The obtained functions form a solution of the bi-decomposition problem.Commutative ideals of BCK-algebras based on uni-hesitant fuzzy set theory.https://www.zbmath.org/1455.060092021-03-30T15:24:00+00:00"Aldhafeeri, Shuaa"https://www.zbmath.org/authors/?q=ai:aldhafeeri.shuaa"Muhiuddin, G."https://www.zbmath.org/authors/?q=ai:muhiuddin.ghulamRelations between uni-hesitant fuzzy commutative ideals and uni-hesitant fuzzy ideals of BCK-algebras are discussed. Conditions for a uni-hesitant
fuzzy ideal to be a uni-hesitant fuzzy commutative ideal are provided.
Extension property for a uni-hesitant fuzzy commutative ideal is
established. A part of results is a consequence of the so-called transfer principle for fuzzy sets.
Reviewer: Wiesław A. Dudek (Wrocław)