Recent zbMATH articles in MSC 06Fhttps://www.zbmath.org/atom/cc/06F2021-03-30T15:24:00+00:00WerkzeugInt-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)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].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)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)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)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)