Recent zbMATH articles in MSC 08https://zbmath.org/atom/cc/082024-02-15T19:53:11.284213ZWerkzeugAdmissible subsets and completions of ordered algebrashttps://zbmath.org/1526.060032024-02-15T19:53:11.284213Z"Laan, Valdis"https://zbmath.org/authors/?q=ai:laan.valdis"Feng, Jianjun"https://zbmath.org/authors/?q=ai:feng.jianjun"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xiaOrdered \(\Omega\)-algebras of fixed type \(\Omega\) are considered. \textit{Linear functions} on an ordered algebra \(\mathcal{A}\) are defined iteratively: 1) identity mapping on \(\mathcal{A}\) is a linear function and 2) if \(p: \mathcal{A}\to\mathcal{A}\) is a linear function and \(\omega\in \Omega_n\), \(a_i\in \mathcal{A}\), then \(x\mapsto \omega(a_1,\ldots, a_{i-1},p(x), a_{i+1},\ldots, a_n)\) defines a linear function as well. Linear functions with \(p=\operatorname{id}_A\) are called \textit{elementary translations}. \(A\) is a \textit{sup-algebra} if it as a poset is a complete lattice and all elementary translations preserve joins. A subset \(M\) of an ordered algebra \(\mathcal{A}\) is called \textit{admissible} if \(\bigvee p(M)\) exists and \(p(\bigvee M)=\bigvee p(M)\) for all \(p\in L_{\mathcal{A}}\), where \( L_{\mathcal{A}}\) is the set of all linear functions on \(\mathcal{A}\). A lower subset \(S\) of an ordered algebra \(\mathcal{A}\) is called a \(\mathcal{D}\)-\textit{ideal} if for any admissible subset \(M\) of \(S\), one has that \(\bigvee M\in S\).
The authors prove that for an ordered algebra \(\mathcal{A}\), for every sup-algebra \(\mathcal{Q}\) and every \texttt{OAlg}\(^*\)-morphism \(f: \mathcal{A}\to \mathcal{Q}\), there exists a unique \texttt{SupAlg}-morphism \(g: \mathcal{D}(A)\to \mathcal{Q}\) such that \(gr=f\), where \(r: \mathcal{A}\to \mathcal{D}(A), a\mapsto a\!\downarrow\) (\texttt{OAlg}\(^*\) is a category of all ordered algebras with admissible-join preserving homomorphisms, \texttt{SupAlg} is a category of all sup-algeras with join-preserving homomorphisms and \(\mathcal{D}(A)\) is the set of all \(\mathcal{D}\)-ideals of \(A\)). It turnes out that \(\mathcal{D}(A)\) is a join-completion of \(\mathcal{A}\) and that \texttt{SupAlg} is a reflective subcategory of \texttt{OAlg}\(^*\).
Reviewer: Peeter Normak (Tallinn)Representations of Menger hypercompositional algebras by some types of commutative hyperoperationshttps://zbmath.org/1526.080012024-02-15T19:53:11.284213Z"Kumduang, Thodsaporn"https://zbmath.org/authors/?q=ai:kumduang.thodsapornMenger algebras were introduced in [\textit{K. Menger}, Rend. Mat. Appl., V. Ser. 20, 409--430 (1961, Zbl 0113.03904)]. Menger hyperalgebras were investigated in [\textit{T. Kumduang} and \textit{S. Leeratanavalee}, Commun. Algebra 49, No. 4, 1513--1533 (2021, Zbl 1472.08002)]. In this paper, the author studies relationship between general Menger hyperalgebras and Menger hyperalgebras of \(k\)-commutative hyperoperations.
Reviewer: Agata Pilitowska (Warszawa)Finitely generated dyadic convex setshttps://zbmath.org/1526.520012024-02-15T19:53:11.284213Z"Matczak, K."https://zbmath.org/authors/?q=ai:matczak.katarzyna"Mućka, A."https://zbmath.org/authors/?q=ai:mucka.anna"Romanowska, A. B."https://zbmath.org/authors/?q=ai:romanowska.anna-bThe paper deals with dyadic rational numbers \(\mathbb{D} =\mathbb{Z}[1/2]\) and the groupoid \((\mathbb{D}^n,\circ)\), where \(a\circ b=(a+b)/2\) for \(a, b\in\mathbb{D}^n\) (arithmetic mean of dyadic vectors).
Dyadic convex sets are defined as subsets of \(\mathbb{D}^n\) closed under the operation \(\circ\). As particular cases, dyadic polygons in \(\mathbb{D}^n, n>1\), and dyadic closed intervals in \(\mathbb{D}\) are considered.
It was known (cf. [the first author et al., Int. J. Algebra Comput. 21, No. 3, 387--408 (2011; Zbl 1223.52008)]) that every dyadic polygon is finitely generated. The paper characterizes all finitely generated dyadic convex sets (Theorem 5.1). Moreover, in Section 6, a method for determining a minimal set of generators and a method for estimating the number of generators are presented in the case of the dyadic plane \(\mathbb{D}^2\) (by examples of dyadic triangles from Proposition 6.2).
Reviewer: Józef Drewniak (Rzeszów)Simulating a quantum composite system by coupled classical oscillatorshttps://zbmath.org/1526.810252024-02-15T19:53:11.284213Z"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jing.12|yang.jing.1|yang.jing.6|yang.jing.5|yang.jing.7|yang.jing.4|yang.jing.2|yang.jingSummary: It is known that the adiabatic evolution and Berry phase of a quantum can be mapped into the classical dynamics and Hannay's angle of coupled oscillators in a strictly mathematical way. In this work, we show that a quantum composite system consisting of two spin-1/2 can also be mapped into coupled two sets of classical oscillators by using this quantum-classical mapping. The evolution and geometric phase for the quantum composite system have been mapped into the classical dynamics and Hannay's angle for the oscillators. The quantum entanglement has also been mapped into the correlation between the two sets of coupled oscillators. We also provide a way to study the classical geometric angle by mapping the geometric phases for the quantum subsystems into classical ones. Our results can provide a new way to simulate the quantum composite system and study the classical composite system.