Recent zbMATH articles in MSC 06 https://www.zbmath.org/atom/cc/06 2022-01-14T13:23:02.489162Z Werkzeug Fixed-point elimination in the intuitionistic propositional calculus https://www.zbmath.org/1475.03069 2022-01-14T13:23:02.489162Z "Ghilardi, Silvio" https://www.zbmath.org/authors/?q=ai:ghilardi.silvio "Gouveia, Maria João" https://www.zbmath.org/authors/?q=ai:gouveia.maria-joao "Santocanale, Luigi" https://www.zbmath.org/authors/?q=ai:santocanale.luigi Summary: It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras -- that is, the algebraic models of the intuitionistic propositional calculus -- always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $$\mu$$-calculus based on intuitionistic logic is trivial, every $$\mu$$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $$\mu$$-formula. The axiomatization of the greatest fixed-pointis simple. The axiomatization of the least fixed-pointis more complex, in particular every monotone formula converges to its least fixed-pointby Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such $$n$$, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal. For the entire collection see [Zbl 1333.68011]. A topological approach to MTL-algebras https://www.zbmath.org/1475.03102 2022-01-14T13:23:02.489162Z "Fussner, Wesley" https://www.zbmath.org/authors/?q=ai:fussner.wesley "Ugolini, Sara" https://www.zbmath.org/authors/?q=ai:ugolini.sara Triples decompostions for algebraic structures ha vea long history. In this paper, the authors provide a dualized construction of Aguzzoli-Flaminio-Ugolini of a large class of MTL-algebras from quadruples ($$B,A, \vee_e,\delta$$). The extended Priestley dual of each $$srDL$$-algebra from the Stone space associated to its Boolean skeleton, the extended Priestley dual of its radical, and a collection of connecting maps, via a rotation construction are constructed. Reviewer: Hongxing Liu (Jinan) Fuzzy UP-ideals and fuzzy UP-subalgebras of UP-algebras in term of level subsets https://www.zbmath.org/1475.03103 2022-01-14T13:23:02.489162Z "Poungsumpao, Polatip" https://www.zbmath.org/authors/?q=ai:poungsumpao.polatip "Kaijae, Waraphorn" https://www.zbmath.org/authors/?q=ai:kaijae.waraphorn "Arayarangsi, Saranya" https://www.zbmath.org/authors/?q=ai:arayarangsi.saranya "Iampan, Aiyared" https://www.zbmath.org/authors/?q=ai:iampan.aiyared The authors consider properties of fuzzy UP-subalgebras and fuzzy UP-ideals of UP-algebras, and prove their basic properties. An algebra $$(A,\cdot, 0)$$ of type $$(2,0)$$ is called a UP-algebra if, for all $$x,y,z\in A$$, (UP-1) $$(y\cdot z)\cdot ((x\cdot y)\cdot (x\cdot z))=0$$, (UP-2) $$0\cdot x = x$$, (UP-3) $$x\cdot 0 = 0$$, (UP-4) $$x\cdot y = y\cdot x = 0$$ implies $$x = y$$. A typical example of UP-algebras is $$(\mathcal{P}(X),\cdot, \emptyset)$$, where $$X$$ is a non-empty set, and $$A\cdot B$$ is defined by $A\cdot B= A-B (= A\cap B^{c}) \ \ \ \ (A,B,\in \mathcal{P}(X)).$ Thus, UP-algebras are closely related to BCK-algebras, BCI-algebras, and so on. A non-empty subset $$B$$ of a UP-algebra $$A$$ is called a UP-ideal if it satisfies \begin{enumerate} \item $$0\in B$$ and \item $$x\cdot (y\cdot z)\in B$$ and $$y\in B$$ imply $$x\cdot z\in B$$. \end{enumerate} A non-empty subset $$B\subseteq A$$ is a UP-subalgebra if $$S$$ is closed under the operation $$\cdot$$''. From these definitions, a fuzzy $$f$$ in $$A$$ is called a fuzzy UP-ideal if, for any $$x, y, z\in A$$, 1. $$f(0)\ge f(x)$$, and 2. $$f(x\cdot z) \ge \min \{f(x\cdot (y\cdot z)), f(y)\}$$. Also, a fuzzy set $$f$$ in $$A$$ is called a fuzzy UP-subalgebra if, for any $$x, y\in A$$, $f(x\cdot y)\ge \min \{f(x), f(y)\}.$As expected, the authors prove the following basic theorems: Theorem 2.1. Every fuzzy UP-ideal is a fuzzy UP-subalgebra. Theorem 2.5, 2.6. Let $$f$$ be a fuzzy set in $$A$$. Then the following statements hold: 1. $$f$$ is a fuzzy UP-ideal (fuzzy UP-subalgebra) if and only if, for all $$t\in [0, 1]$$, $$U(f; t)\neq \emptyset$$ implies $$U(f; t)$$ is a UP-ideal (fuzzy UP-subalgebra), where $$U(f; t) =\{x\in A \mid f(x) \ge t\}$$ 2. $$\bar{f}$$ is a fuzzy UP-ideal (fuzzy UP-subalgebra) if and only if, for all $$t\in [0, 1]$$, $$L(\bar{f}; t)\neq \emptyset$$ implies $$L(\bar{f}; t)$$ is a UP-ideal (fuzzy UP-subalgebra), where $$\bar{f}(x)=1-f(x)$$ and $$L(f; t) = \{x\in A \mid f(x)\le t\}$$. The results above mean that every fuzzy UP-ideal (fuzzy UP-subalgebra) is represented by so-called level-cut UP-ideals (UP-subalgebras) $$U(f; t)$$. Moreover, they prove the following theorems, which roughly mean that an inverse image of a fuzzy UP-ideal (fuzzy UP-subalgebra) by a UP-homomorphism is a fuzzy UP-ideal (fuzzy UP-subalgebra) and an image of a fuzzy UP-ideal (fuzzy UP-subalgebra) is also a fuzzy UP-ideal (fuzzy UP-subalgebra) for a UP-homomorphism satisfying a sup property, that is, for any non-empty $$T$$ of $$A$$, there exists $$T_0\in T$$ such that $$f(t_0) = \sup \{f(t)\}_{t\in T}$$: Theorem 2.14. Let $$(A,\cdot, 0)$$ and $$(B,\cdot,0)$$ be UP-algebras and let $$f : A \to B$$ a UP-epimorphism. If $$\beta$$ is a fuzzy UP-ideal (fuzzy UP-subalgebra) of $$B$$, then $$\beta \circ f$$ is a fuzzy UP-ideal (fuzzy UP-subalgebra) of A. Reviewer: Michiro Kondo (Inzai) Independent sets in tensor products of three vertex-transitive graphs https://www.zbmath.org/1475.05164 2022-01-14T13:23:02.489162Z "Mao, Huiqun" https://www.zbmath.org/authors/?q=ai:mao.huiqun "Zhang, Huajun" https://www.zbmath.org/authors/?q=ai:zhang.huajun Summary: The tensor product $$T(G_1,G_2,G_3)$$ of graphs $$G_1, G_2$$ and $$G_3$$ is defined by $VT(G_1,G_2,G_3) = V(G_1) \times V(G_2) \times V(G_3)$ and $ET(G_1,G_2,G_3) = \{ [(u_1,u_2,u_3), (v_1,v_2,v_3)]: |\{ i: (u_i,v_i) \in E(G_i)\}| \geq 2\}.$ From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of $$T(G_1, G_2, G_3)$$. So $\alpha T(G_1,G_2,G_3) \geq \max\{ \alpha (G_1) \alpha (G_2)|G_3|, \alpha (G_1) \alpha (G_3)|G_2|, \alpha (G_2) \alpha (G_3)|G_1|\}.$ In this paper, we prove that the equality holds if $$G_1$$ and $$G_2$$ are vertex-transitive graphs and $$G_3$$ is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of $$T(G_1, G_2, G_3)$$ is determined. A duality for involutive bisemilattices https://www.zbmath.org/1475.06001 2022-01-14T13:23:02.489162Z "Bonzio, Stefano" https://www.zbmath.org/authors/?q=ai:bonzio.stefano "Loi, Andrea" https://www.zbmath.org/authors/?q=ai:loi.andrea "Peruzzi, Luisa" https://www.zbmath.org/authors/?q=ai:peruzzi.luisa In this paper, the authors provide a duality between the categories of involutive bisemilattices and GR spaces, which are compact totally disconnected partially ordered left normal bands with constants. Involutive bisemilattices are representable as Plonka sums over a direct system of Boolean algebras. Using this, the authors describe the dual spaces of an involutive bisemilattice inverse systems of Stone spaces. They generalize the Gierz and Romanowska duality for GR spaces with involution as an additional operation [\textit{G. Gierz} and \textit{A. Romanowska}, J. Aust. Math. Soc., Ser. A 51, No. 2, 247--275 (1991; Zbl 0751.06008)]. Reviewer: Ágota Figula (Debrecen) $$s_{Z}$$-quasicontinuous posets and meet $$s_{Z}$$-continuous posets https://www.zbmath.org/1475.06002 2022-01-14T13:23:02.489162Z "Ruan, Xiaojun" https://www.zbmath.org/authors/?q=ai:ruan.xiaojun "Xu, Xiaoquan" https://www.zbmath.org/authors/?q=ai:xu.xiaoquan As a common generalization of $$s_2$$-quasicontinuous posets and quasi $$Z$$-continuous domains, the authors introduced the concept of $$s_Z$$-quasicontinuous posets and investigated some of their basic properties. Firstly, they proved that if a subset system $$Z$$ satisfies certain conditions, and $$P$$ is an $$s_z$$-quasicontinuous poset, then the $$Z$$-way below relation $$\ll_Z$$ on $$P$$ has the interpolation property. Then they showed that the space $$(P,\sigma_Z(P))$$ is locally compact and the space $$(P,\lambda_Z(P))$$ is a pospace. Finally, they proved that under some conditions, a poset is $$s_Z$$-continuous if and only if it is meet $$s_Z$$-continuous and $$s_Z$$-quasicontinuous. These results are interesting and extend the framework of domain theory. Reviewer: Wenfeng Zhang (Nanchang) The lattice of subspaces of a vector space over a finite field https://www.zbmath.org/1475.06003 2022-01-14T13:23:02.489162Z "Chajda, Ivan" https://www.zbmath.org/authors/?q=ai:chajda.ivan "Länger, Helmut" https://www.zbmath.org/authors/?q=ai:langer.helmut-m The lattice of subspaces of a finite dimensional Hilbert space over a finite field is orthocomplemented if and only if the finite field has characteristic $$2$$ and the vector space dimension $$2$$ [\textit{J. P. Eckman} and \textit{P. C. Zabey}, Helv. Phys. Acta 42, 420--424 (1969; Zbl 0181.56601)]. The purpose of this paper is to show: with respect to a suitable choice of complementation on the lattice of subspaces of a finite dimensional vector space over a finite field, the lattice is weakly orthomodular and dually weakly orthomododular in the sense of the authors [Order 35, No. 3, 541--555 (2018; Zbl 1441.06005)] (Corollary 12). Furthermore lattices of subspaces of finite dimensional vector spaces over finite fields are shown to be paraorthomodular (in the sense of \textit{R. Giuntini} et al. [Stud. Log. 104, No. 6, 1145--1177 (2016; Zbl 1417.06008)], or Definition 19 of the paper) with respect to orthogonality (Proposition 20 and Corollary 21). Reviewer: Partha Ghosh (Johannesburg) Single identities forcing lattices to be Boolean https://www.zbmath.org/1475.06004 2022-01-14T13:23:02.489162Z "Chajda, Ivan" https://www.zbmath.org/authors/?q=ai:chajda.ivan "Länger, Helmut" https://www.zbmath.org/authors/?q=ai:langer.helmut-m "Padmanabhan, Ranganathan" https://www.zbmath.org/authors/?q=ai:padmanabhan.ranganathan Summary: In this note we characterize Boolean algebras among lattices of type (2, 2, 1) with join, meet and an additional unary operation by means of single two-variable respectively three-variable identities. In particular, any uniquely complemented lattice satisfying any one of these equational constraints is distributive and hence a Boolean algebra. Pseudocompact frames $$L$$ versus different topologies on $$R(L)$$ https://www.zbmath.org/1475.06005 2022-01-14T13:23:02.489162Z "Acharyya, Sudip Kumar" https://www.zbmath.org/authors/?q=ai:acharyya.sudip-kumar "Bhunia, Goutam" https://www.zbmath.org/authors/?q=ai:bhunia.goutam "Ghosh, Partha Pratim" https://www.zbmath.org/authors/?q=ai:ghosh.partha-pratim Summary: In this paper we have characterized pseudocompact frames $$L$$ (1) via $$u$$-topology and $$m$$-topology on the rings $$R(L)$$ and $$R^\ast(L)$$; (2) via some special kind of ideals of $$\mathrm{Coz}\,L$$. Free modal pseudocomplemented De Morgan algebras https://www.zbmath.org/1475.06006 2022-01-14T13:23:02.489162Z "Figallo, Aldo V." https://www.zbmath.org/authors/?q=ai:figallo.aldo-victorio "Oliva, Nora" https://www.zbmath.org/authors/?q=ai:oliva.nora "Ziliani, Alicia" https://www.zbmath.org/authors/?q=ai:ziliani.alicia Summary: Modal pseudocomplemented De Morgan algebras (or \textit{mpM}-algebras) were investigated in [\textit{A. V. Figallo} et al., Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 53, No. 1, 65--79 (2014; Zbl 1347.06011)], and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying $$x\wedge (\sim x)^*=(\sim (x\wedge (\sim x)^*))^*$$ studied by \textit{H. P. Sankappanavar} [Z. Math. Logik Grundlagen Math. 33, 3--11 (1987; Zbl 0624.06016)]. In this paper the study of these algebras is continued. More precisely, new characterizations of \textit{mpM}-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the fi nite \textit{mpM}-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated \textit{mpM}-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [\textit{Yu. M. Movsisyan} and \textit{V. A. Aslanyan}, Commun. Algebra 42, No. 11, 4757--4777 (2014; Zbl 1338.06010); Cent. Eur. J. Math. 12, No. 12, 1749--1761 (2014; Zbl 1346.08004); Demonstr. Math. 47, No. 2, 271--283 (2014; Zbl 1315.06012)]. Convex MV-algebras: many-valued logics meet decision theory https://www.zbmath.org/1475.06007 2022-01-14T13:23:02.489162Z "Flaminio, T." https://www.zbmath.org/authors/?q=ai:flaminio.tommaso "Hosni, H." https://www.zbmath.org/authors/?q=ai:hosni.hykel "Lapenta, S." https://www.zbmath.org/authors/?q=ai:lapenta.serafina In order to have logico-algebraic instruments in the framework of decision theory under uncertainty, the authors introduce the notion of convex combination in MV-algebras. In particular they define algebraic structures obtained by adding to MV-algebras a family of binary operations, indexed by real numbers $$\alpha \in [0,1]$$, playing the role of convex combination of two points with parameter $$\alpha$$. The obtained class of algebraic structures, called MV-algebras with convexity operators, are shown to be term equivalent to Riesz MV-algebras. Some results for such class are proved, as for example that no countable MV-algebra can be equipped with a family of convexity operators. Finally, MV-algebras with convexity operators are shown to be useful to approach the foundations of decision theory under uncertainty, in particula to give a logico-algebraic representation of the Anscombe-Aumann problem. Reviewer: Brunella Gerla (Varese) Int-soft ideals of pseudo MV-algebras https://www.zbmath.org/1475.06008 2022-01-14T13:23:02.489162Z "Jun, Young B." https://www.zbmath.org/authors/?q=ai:jun.young-bae "Song, Seok-Zun" https://www.zbmath.org/authors/?q=ai:song.seok-zun "Bordbar, Hashem" https://www.zbmath.org/authors/?q=ai:bordbar.hashem After that Chang proved the completeness of Lukasiewicz logic and introduced the notion of MV-algebra (MV for many-valued), many researchers have studied and investigated the properties of this structure and various of its generalizations. One of these generalizations is the non-commutative case in which the multiplication operation (t-norm) is non-commutative. This new structure is introduced separately [\textit{J. Rachůnek}, Czech. Math. J. 52, No. 2, 255--273 (2002; Zbl 1012.06012); \textit{G. Georgescu} and \textit{A. Iorgulescu}, Mult.-Valued Log. 6, No. 1--2, 95--135 (2001; Zbl 1014.06008)]. On the other hand, there are several ways to modeling uncertain events. One of them is introduced by \textit{D. Molodtsov} [Comput. Math. Appl. 37, No. 4--5, 19--31 (1999; Zbl 0936.03049)]. In this paper, the authors apply soft set theory on pseudo MV-algebras and investigate the properties of substructures. To do this, they introduce the notion of int-soft ideals, and give some properties and characterizations of them. Another useful reference is [\textit{A. Dvurečenskij}, Soft Comput. 5, No. 5, 347--354 (2001; Zbl 0998.06010)]. Reviewer: Mahmood Bakhshi (Bojnord) New types of $(\alpha,\beta)$-fuzzy subalgebras of BCK/BCI-algebras https://www.zbmath.org/1475.06009 2022-01-14T13:23:02.489162Z "Muhiuddin, G." https://www.zbmath.org/authors/?q=ai:muhiuddin.ghulam "Shum, K. P." https://www.zbmath.org/authors/?q=ai:shum.kar-ping The notions of ($$\alpha,\beta$$)-fuzzy subalgebras of BCK\slash BCI-algebras are introduced in this paper. Also, the authors provide some conditions for a subset to be a subalgebra of BCK/BCI-algebras. Reviewer: Hongxing Liu (Jinan) A study on pseudoorders in ordered $$\ast$$-semihypergroups https://www.zbmath.org/1475.06010 2022-01-14T13:23:02.489162Z "Feng, Xinyang" https://www.zbmath.org/authors/?q=ai:feng.xinyang "Tang, Jian" https://www.zbmath.org/authors/?q=ai:tang.jian "Luo, Yanfeng" https://www.zbmath.org/authors/?q=ai:luo.yan-feng In this paper, the authors study the pseudoorders on ordered $$\ast$$-semihypergroups in detail. To begin with, the authors introduce pseudoorders on an ordered $$\ast$$-semihypergroup and investigate their related properties. Furthermore, the relationship between strongly regular equivalence relations and pseudoorders on an ordered $$\ast$$-semihypergroup is established and some homomorphism theorems of ordered $$\ast$$-semihypergroups by pseudoorders are given. Finally, the authors investigate the direct product of ordered $$\ast$$-semihypergroups and study the pseudoorders on direct product of ordered $$\ast$$-semihypergroups. Reviewer: Xie Xiang-Yun (Guangdong) A graphical calculus for integration over random diagonal unitary matrices https://www.zbmath.org/1475.15047 2022-01-14T13:23:02.489162Z "Nechita, Ion" https://www.zbmath.org/authors/?q=ai:nechita.ion "Singh, Satvik" https://www.zbmath.org/authors/?q=ai:singh.satvik The authors present a graphical tool to compute the expectation of tensor network diagrams containing two collections of uniformly distributed unitary random vectors on a unit circle. The relevant mathematical tools, both from combinatorics and from graphical expressions for quantum networks, are briefly given in Section 3. The main result is stated in Theorem 4.8 where it is shown the the expectation can be expressed as a summation over diagrams constrained by several different rules of connections among the links replacing the random vectors. A similar statement is presented for the case of real vectors in Theorem 5.5. The remaining parts of the article exploit the applications of the two main relations to bipartite system networks concerning random unitary matrices including the analysis of twirling maps between matrix algebras. In addition, some observations for tripartite system networks are discussed in the application sections. The paper is written with heavy technicality in mathematics, hence the readers must be familiar with the concepts and the nomenclature used in combinatorics to well understand the analysis in the paper. However, the results and their derivations are well organised and straightforward. For the readers who are not familiar with combinatorics, I suggest, to begin with, the main statement in Section 4, especially with Theorem 4.8. The mathematical background in Section 3 can be used to follow the proof. In the application sections, despite the detailed discussions for the employed examples, there remain several open problems that one can investigate both mathematically and physically. For instance, the generalisation to the networks concerning multipartite matrices, the consideration of the different types of the underlying distribution, or the connection to the physical implementation of the considered networks, remain open. Reviewer: Fattah Sakuldee (Warszawa) On compactness and cocompactness conditions for $$R$$-\textit{tors}, $$R$$-\textit{pr} and (big) lattices of classes of modules https://www.zbmath.org/1475.16013 2022-01-14T13:23:02.489162Z "Rincón-Mejía, Hugo Alberto" https://www.zbmath.org/authors/?q=ai:rincon-mejia.hugo-alberto "Sánchez-Hernández, José Patricio" https://www.zbmath.org/authors/?q=ai:sanchez-hernandez.jose-patricio "Sandoval-Miranda, Martha Lizbeth Shaid" https://www.zbmath.org/authors/?q=ai:sandoval-miranda.martha-lizbeth-shaid "Zorrilla-Noriega, Manuel" https://www.zbmath.org/authors/?q=ai:zorrilla-noriega.manuel-gerardo Summary: We study conditions related to compactness and cocompactness for some (big) lattices of classes of modules and preradicals. Also, we give some characterizations in terms of rings and modules. The Möbius function of $$\mathrm{PSU}(3, 2^{2^n})$$ https://www.zbmath.org/1475.20082 2022-01-14T13:23:02.489162Z "Zini, Giovanni" https://www.zbmath.org/authors/?q=ai:zini.giovanni Let $$G$$ be a finite group and $$H$$ be a subgroup of $$G$$. The Möbius function $$\mu(H,G)$$ is defined as $$\mu(G,G)=1$$ and $$\sum_{K\geq H}\mu(K,G)=0$$ if $$H<G$$. The Möbius function is defined more generally on a locally finite poset $$(\mathcal{P},\leq)$$ as $$\mu(x,x)=1$$, $$\mu(x,y)=0$$ if $$x\nleq y$$, and $$\sum_{x\leq z\leq y}\mu(z,y)=0$$ if $$x\leq y$$. It is denoted by $$\lambda(H,G)$$ the Möbius function $$\mu([H],[G])$$ in the poset $$\bar{L}$$ of conjugacy classes $$[H]$$ of subgroups $$H$$ in $$G$$, ordered as: $$[H]\leq [K]$$ if and only if $$H$$ is contained in some conjugate of $$K$$ in $$G$$. In this paper, the author considers the three dimensional projective special unitary group $$G=\mathrm{PSU}(3,q)$$ over the field with $$q=2^{2^n}$$ elements, for any $$n\geq 1$$. $$\mu(H,G)$$ is computed for any subgroup $$H$$ of $$G$$, which shows that the groups $$\mathrm{PSU}(3,2^{2^n})$$ satisfy \textit{A. Mann}'s conjecture discussed in [Forum Math. 8, 429--459 (1996; Zbl 0852.20019)]. Moreover, $$\lambda(H,G)$$ is computed for any subgroup $$H$$ of $$G$$, which shows that the groups $$\mathrm{PSU}(3,2^{2^n})$$ satisfy the $$(\mu,\lambda)$$-property (i.e. if $$G$$ is solvable, then $$\mu(\{1\},G)=|G'|\cdot \lambda(\{1\},G)$$), but do not satisfy the generalized $$(\mu,\lambda)$$-property (i.e. if $$G$$ is solvable, then $$\mu(H,G)=[N_{G'}(H):H\cap G']\cdot\lambda(H,G)$$, for any subgroup $$H$$ of $$G$$). Finally, the Euler characteristic $$\chi(\Delta(L_p\backslash \{1\}))$$ of the order complex of the poset $$L_p\backslash \{1\}$$ of non-trivial $$p$$-subgroups of $$G$$ is computed for any prime $$p$$. Reviewer: Hesam Safa (Bojnord) On almost-symmetry in generalized numerical semigroups https://www.zbmath.org/1475.20091 2022-01-14T13:23:02.489162Z "Cisto, Carmelo" https://www.zbmath.org/authors/?q=ai:cisto.carmelo "Tenório, Wanderson" https://www.zbmath.org/authors/?q=ai:tenorio.wanderson A generalized numerical semigroup (GNS) is a submonoid, $$S\subset\mathbb{N}^d$$, $$d\in\mathbb{N}$$, such that $$|H(S)|=|\mathbb{N}^d\setminus S|<\infty$$. The set $$H(S)$$ is called the set of gaps and its cardinality is the genus of $$S$$. The elements $$h\in H(S)$$ such that $$h+s\in S$$ for all $$s\in S\setminus\{0\}$$ are called the pseudo-Frobenius elements and the set of all pseudo-Frobenius elements of $$S$$ is denoted by $$\mathrm{PF}(S)$$. The number $$|\mathrm{PF}(S)|$$ is the type of $$S$$. In this paper, a family of monoids is introduced, the almost symmetric GNS, using the genus, the type and the set of gaps of $$S$$. This family is studied and a characterization of pseudo-symmetric GNS is given using the almost symmetric GNS. The authors also give a characterization of almost symmetric GNS using the reduced Apéry set and they count all almost symmetric GNS with a fixed Frobenius element. Reviewer: Daniel Marín Aragon (Cádiz) On images of complete topologized subsemilattices in sequential semitopological semilattices https://www.zbmath.org/1475.22007 2022-01-14T13:23:02.489162Z "Banakh, Taras" https://www.zbmath.org/authors/?q=ai:banakh.taras-o "Bardyla, Serhii" https://www.zbmath.org/authors/?q=ai:bardyla.serhii A topologized semilattice $$X$$ (i.e., $$X$$ is a semilattice endowed with a topology) is called a (\textit{semitopological semilattice} if the semilattice operation $$\wedge : X\times X \rightarrow X, (x, y)\mapsto xy$$, is (separately) continuous. A topologized semilattice $$X$$ is called \textit{complete} if each non-empty chain $$C\subseteq X$$ has $$\inf C\in \overline {C}$$ and $$\inf C\in \overline {C}$$. A topological space $$Y$$ is said to be \textit{sequential} if for any subset $$A$$ of $$Y$$, $$A$$ is closed iff $$A$$ is sequentially closed (that is, $$A$$ contains the limits of all sequences $$(a_n)_{n\in \omega}$$ in $$A$$ that converge in $$Y$$). A poset $$P$$ is called \textit{chain-finite} if each chain in $$P$$ is finite. In 1975, Stepp proved that for any homomorphism $$h: X \rightarrow Y$$ from a chain-finite semilattice to a Hausdorff topological semilattice $$Y$$ the image $$h(X)$$ is closed in $$Y$$. In 2019, this result was improved by Banakh and Bardyla by showing that the result holds for $$Y$$ being a Hausdorff semitopological semilattice. They also proved that for any continuous homomorphism $$h : X \rightarrow Y$$ from a complete topologized semilattice $$X$$ to a Hausdorff topological semilattice $$Y$$ the image $$h(X)$$ is closed in $$Y$$. In this paper, the authors continue such investigations. It is proved that for any continuous homomorphism $$h: X \rightarrow Y$$ from a complete topologized semilattice $$X$$ to a sequential Hausdorff semitopological semilattice $$Y$$ the image $$h(X)$$ is closed in $$Y$$. Reviewer: Xiaoquan Xu (Zhangzhou) Removahedral congruences versus permutree congruences https://www.zbmath.org/1475.52017 2022-01-14T13:23:02.489162Z "Albertin, Doriann" https://www.zbmath.org/authors/?q=ai:albertin.doriann "Pilaud, Vincent" https://www.zbmath.org/authors/?q=ai:pilaud.vincent "Ritter, Julian" https://www.zbmath.org/authors/?q=ai:ritter.julian Summary: The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cones of the permutree fans. Two types of Galois correspondences over quantaloid-typed sets https://www.zbmath.org/1475.54005 2022-01-14T13:23:02.489162Z "Fang, Jinming" https://www.zbmath.org/authors/?q=ai:fang.jinming "Fang, Zhou" https://www.zbmath.org/authors/?q=ai:fang.zhou There exist the so-called \textit{Lowen functors}, which provide a correspondence between the categories of crisp and fuzzy topological spaces [\textit{R. Lowen}, J. Math. Anal. Appl. 56, 621--633 (1976; Zbl 0342.54003)]. These functors have already been generalized in several ways (see, e.g., [\textit{U. Höhle}, Many valued topology and its applications. Boston, MA: Kluwer Academic Publishers (2001; Zbl 0969.54002); \textit{U. Höhle} and \textit{T. Kubiak}, Semigroup Forum 75, No. 1, 1--17 (2007; Zbl 1125.06006); \textit{D. Zhang}, Fuzzy Sets Syst. 140, No. 3, 479--487 (2003; Zbl 1086.54502)]), either replacing the category of topological spaces with the category of, e.g., limit spaces or employing a certain lattice-theoretic structure instead of the unit interval underlying fuzzy topological spaces. The present paper follows suit and provides yet another extension of Lowen functors. The authors combine the above-mentioned generalizations by, first, replacing the underlying lattice $$L$$ of lattice-valued topology by a quantaloid $$\mathcal{Q}$$ (see, e.g., [\textit{K. I. Rosenthal}, The theory of quantaloids. Harlow: Addison Wesley Longman (1996; Zbl 0845.18003)] for more details on quantaloids); and, second, replacing the category of topological spaces by the category of suitably defined limit spaces. Thus, instead of lattice-valued sets, the authors rely on quantaloid-typed sets. Moreover, Lowen functors are represented in the form of a Galois correspondence between concrete categories, employing the language of [\textit{J. Adámek} et al., Repr. Theory Appl. Categ. 2006, No. 17, 1--507 (2006; Zbl 1113.18001)]. The authors additionally arrive at a quantaloid-enriched extension of the notion of $$\top$$-filter of \textit{U. Höhle} [Manuscr. Math. 38, 289--323 (1982; Zbl 1004.54500)], which leads to the concept of $$\top$$-limit space as well as a Galois correspondence between the categories of quantaloid-enriched limit spaces and quantaloid-enriched $$\top$$-limit spaces. The paper is well written, provides most of its required preliminaries, but is quite technical and thus will require a bit of patience from the reader (supposedly a categorical fuzzy topologist), who would like to dwell into all its subtleties. Reviewer: Sergejs Solovjovs (Praha) First-countability, $$\omega$$-Rudin spaces and well-filtered determined spaces https://www.zbmath.org/1475.54012 2022-01-14T13:23:02.489162Z "Xu, Xiaoquan" https://www.zbmath.org/authors/?q=ai:xu.xiaoquan "Shen, Chong" https://www.zbmath.org/authors/?q=ai:shen.chong "Xi, Xiaoyong" https://www.zbmath.org/authors/?q=ai:xi.xiaoyong "Zhao, Dongsheng" https://www.zbmath.org/authors/?q=ai:zhao.dongsheng All spaces are assumed $$T_0$$. Connections involving countable versions, in some sense, of properties such as Rudin, well-filtered and $$d$$-spaces are considered. Every space with first countable sobrification is $$\omega$$-Rudin and if, further, it is $$\omega$$-well-filtered then it is sober. For a first countable space $$X$$ the following four conditions are equivalent: $$X$$ is sober; $$X$$ is well-filtered; $$X$$ is an $$\omega$$-well-filtered $$d$$-space; $$X$$ is an $$\omega$$-well-filtered $$\omega^*$$-$$d$$-space. If $$f:X\to Y$$ is continuous, $$X$$ first countable and $$Y$$ is $$\omega$$-well-filtered, and $$A\subset X$$ irreducible then $$\overline{f(A)}$$ is a Rudin set. Several examples are presented contradicting possible implications involving some other pairs of these and related properties. Reviewer: David B. Gauld (Auckland) Left orderable surgeries of double twist knots. II https://www.zbmath.org/1475.57011 2022-01-14T13:23:02.489162Z "The Khoi, Vu" https://www.zbmath.org/authors/?q=ai:khoi.vu-the "Teragaito, Masakazu" https://www.zbmath.org/authors/?q=ai:teragaito.masakazu "Tran, Anh T." https://www.zbmath.org/authors/?q=ai:tran.anh-tuan Summary: A slope $$r$$ is called a left orderable slope of a knot $$K \subset S^3$$ if the 3-manifold obtained by $$r$$-surgery along $$K$$ has left orderable fundamental group. Consider double twist knots $$C(2m, \pm 2n)$$ and $$C(2m+1, -2n)$$ in the Conway notation, where $$m \ge 1$$ and $$n \ge 2$$ are integers. By using \textit{continuous} families of hyperbolic $$\text{SL}_2(\mathbb{R})$$-representations of knot groups, it was shown in [\textit{R. Hakamata} and \textit{M. Teragaito}, Can. Math. Bull. 57, No. 2, 310--317 (2014; Zbl 1305.57010), \textit{A. T. Tran}, J. Math. Soc. Japan 67, No. 1, 319--338 (2015; Zbl 1419.57028)] that any slope in $$(-4n, 4m)$$ (resp. $$[0, \max\{4m, 4n\})$$) is a left orderable slope of $$C(2m, 2n)$$ (resp. $$C(2m, -2n)$$) and in [\textit{X. Gao}, Slope of orderable Dehn filling of two-bridge knots'', Preprint, \url{arXiv:1912.07468}] that any slope in $$(-4n,0]$$ is a left orderable slope of $$C(2m+1,-2n)$$. However, the proofs of these results are incomplete, since the continuity of the families of representations was not proved. In this paper, we complete these proofs, and, moreover, we show that any slope in $$(-4n, 4m)$$ is a left orderable slope of $$C(2m+1,-2n)$$ detected by hyperbolic $$\text{SL}_2(\mathbb{R})$$-representations of the knot group. For part I, see [\textit{A. T. Tran}, J. Math. Soc. Japan 73, No. 3, 753--765 (2021; Zbl 07398025)]. Rota's Fubini lectures: the first problem https://www.zbmath.org/1475.60006 2022-01-14T13:23:02.489162Z "Mundici, Daniele" https://www.zbmath.org/authors/?q=ai:mundici.daniele Summary: In his 1998 Fubini Lectures, \textit{G. C. Rota} [in: Algebraic combinatorics and computer science. A tribute to Gian-Carlo Rota. Milano: Springer. 57--93 (2001; Zbl 0984.60007)] discusses twelve problems in probability that no one likes to bring up''. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a pointless'' foundation of probability should be provided by algebras of random variables. In 1958 \textit{C. C. Chang} [Trans. Am. Math. Soc. 88, 467--490 (1958; Zbl 0084.00704)] introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic $$\text{Ł}_\infty$$. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem. The adjunction between MV-algebras and unital commutative $$\mathrm{C}^*$$-algebras equips every MV-algebra $$A$$ with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set $$\mathsf{S}(A) \subseteq [0 , 1]^A$$ of finitely additive probability measures on $$A$$ (the states of $$A)$$ coincides with the set of $$[0 , 1]$$-valued functions on $$A$$ whose finite restrictions are consistent in de Finetti's sense. MV-algebras and $$\text{Ł}_\infty$$ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism $$\gamma_A$$ of $$\mathsf{S}(A)$$ onto the weakly compact space of regular Borel probability measures on the maximal spectral space $$\boldsymbol{\mu}(A)$$. The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space $$(\Omega, \mathcal{F}_{\Omega}, P)$$, with $$\mathcal{F}_{\Omega}$$ the sigma-algebra of Borel sets of a compact Hausdorff space $$\Omega$$, and $$P$$ a regular probability measure on $$\mathcal{F}_{\Omega}$$, there is an MV-algebra $$A$$ and a state $$\sigma$$ of $$A$$ such that $$(\Omega, \mathcal{F}_{\Omega}, P)\cong(\boldsymbol{\mu}(A), \mathcal{F}_{\boldsymbol{\mu} ( A )}, \gamma_A(\sigma))$$. Generalized Fréchet bounds for cell entries in multidimensional contingency tables https://www.zbmath.org/1475.62173 2022-01-14T13:23:02.489162Z "Uhler, Caroline" https://www.zbmath.org/authors/?q=ai:uhler.caroline "Richards, Donald" https://www.zbmath.org/authors/?q=ai:richards.donald-l|richards.donald-st-p Summary: We consider the lattice, $$\mathcal{L}$$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $$n(\cdot)$$, on $$\mathcal{L}$$. We derive from the supermodularity of $$n(\cdot)$$ some generalized Fréchet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $$n(\cdot)$$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fréchet inequalities for multidimensional contingency tables. Sensitivity versus certificate complexity of Boolean functions https://www.zbmath.org/1475.68135 2022-01-14T13:23:02.489162Z "Ambainis, Andris" https://www.zbmath.org/authors/?q=ai:ambainis.andris "Prūsis, Krišjānis" https://www.zbmath.org/authors/?q=ai:prusis.krisjanis "Vihrovs, Jevgēnijs" https://www.zbmath.org/authors/?q=ai:vihrovs.jevgenijs Summary: Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that the certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $$\mathit{bs}(f) \leq C(f) \leq 2^{s(f)-1} s(f) - (s(f)-1)$$. In this work, we give a better upper bound of $$\mathit{bs}(f) \leq C(f) \leq \max \left( 2^{s(f)-1}\left( s(f)-\frac{1}{3}\right) , s(f)\right)$$ using a recent theorem limiting the structure of function graphs. We also examine relations between these measures for functions with 1-sensitivity $$s_1(f)=2$$ and arbitrary 0-sensitivity $$s_0(f)$$. For the entire collection see [Zbl 1337.68014]. The next whisky bar https://www.zbmath.org/1475.68136 2022-01-14T13:23:02.489162Z "Behrisch, Mike" https://www.zbmath.org/authors/?q=ai:behrisch.mike "Hermann, Miki" https://www.zbmath.org/authors/?q=ai:hermann.miki "Mengel, Stefan" https://www.zbmath.org/authors/?q=ai:mengel.stefan "Salzer, Gernot" https://www.zbmath.org/authors/?q=ai:salzer.gernot Summary: We determine the complexity of an optimization problem related to information theory. Taking a conjunctive propositional formula over some finite set of Boolean relations as input, we seek a satisfying assignment of the formula having minimal Hamming distance to a given assignment that is not required to be a model (NearestSolution, NSol). We obtain a complete classification with respect to the relations admitted in the formula. For two classes of constraint languages we present polynomial time algorithms; otherwise, we prove hardness or completeness concerning the classes APX, poly-APX, NPO, or equivalence to well-known hard optimization problems. For the entire collection see [Zbl 1337.68014]. The Boolean algebra of piecewise testable languages https://www.zbmath.org/1475.68195 2022-01-14T13:23:02.489162Z "Konovalov, Anton" https://www.zbmath.org/authors/?q=ai:konovalov.anton "Selivanov, Victor" https://www.zbmath.org/authors/?q=ai:selivanov.victor-l Summary: We characterize up to isomorphism the Boolean algebra (BA, for short) of regular piecewise testable languages and show the decidability of classes of regular languages related to this characterization. This BA turns out isomorphic to several other natural BAs of regular languages, in particular to the BA of regular aperiodic languages. For the entire collection see [Zbl 1337.68005]. Describing hierarchy of concept lattice by using matrix https://www.zbmath.org/1475.68370 2022-01-14T13:23:02.489162Z "Pak, Chol Hong" https://www.zbmath.org/authors/?q=ai:pak.chol-hong "Kim, Jin Hong" https://www.zbmath.org/authors/?q=ai:kim.jin-hong|kim.jinhong "Jong, Myong Guk" https://www.zbmath.org/authors/?q=ai:jong.myong-guk Summary: Concept lattices (also called Galois lattices) are complete ones with the hierarchical order relation of the formal concepts defined by a formal context or Galois connection. In this paper, we present a new of method describing a hierarchy of a finite concept lattice by using a matrix. Given a finite concept lattice $$L$$, we introduce Scott topology $$\sigma(L)$$ on $$L$$ and choose an order of a unique minimal base for $$\sigma(L)$$. Then, there is a one-to-one correspondence between the finite topological space $$(L, \sigma(L))$$ and a proper square matrix with integral entries; thus we obtain a hierarchy-matrix describing the hierarchy of the concept lattice. We explain how to get the information of the hierarchy from the hierarchy-matrix and discuss the relation between the hierarchy-matrix and the Hasse diagram. Since the hierarchy-matrix allowed us to store the information of hierarchy of the concept lattice, we believe that any software autonomously understand the information of hierarchy of the concepts from the hierarchy-matrix. Modeling imprecise and bipolar algebraic and topological relations using morphological dilations https://www.zbmath.org/1475.68379 2022-01-14T13:23:02.489162Z "Bloch, Isabelle" https://www.zbmath.org/authors/?q=ai:bloch.isabelle Summary: In many domains of information processing, such as knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, spatial reasoning, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools, such as fusion and aggregation, similarity and distances, mathematical morphology. Here we propose to extend these tools by defining algebraic and topological relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators. These definitions are shown to have the desired properties and to be consistent with existing definitions on sets and fuzzy sets, while providing an additional bipolar feature. The proposed relations can be used for instance for preference modeling or spatial reasoning. They apply more generally to any type of functions taking values in a poset or a complete lattice, such as L-fuzzy sets. Mathematical morphology based on stochastic permutation orderings https://www.zbmath.org/1475.68412 2022-01-14T13:23:02.489162Z "Lézoray, Olivier" https://www.zbmath.org/authors/?q=ai:lezoray.olivier Summary: The extension of mathematical morphology to multivariate data has been an active research topic in recent years. In this paper we propose an approach that relies on the consensus combination of several stochastic permutation orderings. The latter are obtained by searching for a smooth shortest path on a graph representing an image. This path is obtained with a randomized version nearest of neighbors heuristics on a graph. The construction of the graph is of crucial importance and can be based on both spatial and spectral information to enable the obtaining of smoother shortest paths. The starting vertex of a path being taken at random, many different permutation orderings can be obtained and we propose to build a consensus ordering from several permutation orderings. We show the interest of the approach with both quantitative and qualitative results. Flat morphological operators from non-increasing set operators. I: General theory https://www.zbmath.org/1475.68426 2022-01-14T13:23:02.489162Z "Ronse, Christian" https://www.zbmath.org/authors/?q=ai:ronse.christian Summary: Flat morphology is a general method for obtaining increasing operators on grey-level or multivalued images from increasing operators on binary images (or sets). It relies on threshold stacking and superposition; equivalently, Boolean max and min operations are replaced by lattice-theoretical sup and inf operations. In this paper we consider the construction a flat operator on grey-level or colour images from an operator on binary images that is not increasing. Here grey-level and colour images are functions from a space to an interval in $$\mathbb{R}^m$$ or $$\mathbb{Z}^m$$ $$(m \geq 1)$$. Two approaches are proposed. First, we can replace threshold superposition by threshold summation. Next, we can decompose the non-increasing operator on binary images into a linear combination of increasing operators, then apply this linear combination to their flat extensions. Both methods require the operator to have bounded variation, and then both give the same result, which conforms to intuition. Our approach is very general, it can be applied to linear combinations of flat operators, or to linear convolution filters. Our work is based on a mathematical theory of summation of real-valued functions of one variable ranging in a poset. In a second paper, we will study some particular properties of non-increasing flat operators. Unit representation of semiorders. I: Countable sets https://www.zbmath.org/1475.91262 2022-01-14T13:23:02.489162Z "Bouyssou, Denis" https://www.zbmath.org/authors/?q=ai:bouyssou.denis "Pirlot, Marc" https://www.zbmath.org/authors/?q=ai:pirlot.marc Summary: This paper proposes a new proof of the existence of constant threshold representations of semiorders on countably infinite sets. The construction treats each indifference-connected component of the semiorder separately. It uses a partition of such an indifference-connected component into indifference classes. Each element in the indifference-connected component is mirrored, using a ghost'' element, into a reference indifference class that is weakly ordered. A numerical representation of this weak order is used as the basis for the construction of the unit representation after an appropriate lifting operation. We apply the procedure to each indifference-connected component and assemble them adequately to obtain an overall unit representation. Our proof technique has several original features. It uses elementary tools and can be seen as the extension of a technique designed for the finite case, using a denumerable set of inductions. Moreover, it gives us much control on the representation that is built, so that it is, for example, easy to investigate its uniqueness. Finally, we show in a companion paper that our technique can be extended to the general (uncountable) case, almost without changes, through the addition of adequate order-denseness conditions. Unit representation of semiorders. II: The general case https://www.zbmath.org/1475.91263 2022-01-14T13:23:02.489162Z "Bouyssou, Denis" https://www.zbmath.org/authors/?q=ai:bouyssou.denis "Pirlot, Marc" https://www.zbmath.org/authors/?q=ai:pirlot.marc Summary: Necessary and sufficient conditions under which semiorders on uncountable sets can be represented by a real-valued function and a constant threshold are known. We show that the proof strategy that we used for constructing representations in the case of denumerable semiorders can be adapted to the uncountable case. We use it to give an alternative proof of the existence of strict unit representations. In contrast to the countable case, semiorders on uncountable sets that admit a strict unit representation do not necessarily admit a nonstrict unit representation, and conversely. By adapting the proof strategy used for strict unit representations, we establish a characterization of the semiorders that admit a nonstrict representation. Conditions for the existence of other special unit representations are also obtained. For Part I, see [the authors, ibid. 103, Article ID 102566, 21 p. (2021; Zbl )]. Permutation lattices of equivalence relations on the Cartesian products and systems of equations concordant with these lattices. I https://www.zbmath.org/1475.94153 2022-01-14T13:23:02.489162Z "Polin, S. V." https://www.zbmath.org/authors/?q=ai:polin.sergey-v Summary: A description of $$GA$$-lattices previously introduced by the author is given and easily solved systems of equations concordant with these lattices are presented. Permutation lattices of equivalence relations on the Cartesian products and systems of equations concordant with these lattices. II https://www.zbmath.org/1475.94154 2022-01-14T13:23:02.489162Z "Polin, S. V." https://www.zbmath.org/authors/?q=ai:polin.sergey-v Summary: A description of GA-lattices previously introduced by the author is given and easily solved systems of equations concordant with these lattices are presented. For Part I see [the author, ibid. 6, No. 1, 135--158 (2015; Zbl 1475.94153)].