Recent zbMATH articles in MSC 06Bhttps://zbmath.org/atom/cc/06B2024-07-25T18:28:20.333415ZWerkzeugResiduated implications derived from quasi-overlap functions on latticeshttps://zbmath.org/1537.030792024-07-25T18:28:20.333415Z"Paiva, Rui"https://zbmath.org/authors/?q=ai:paiva.rui-c|paiva.rui-pedro"Bedregal, Benjamín"https://zbmath.org/authors/?q=ai:bedregal.benjamin-rene-callejas"Santiago, Regivan"https://zbmath.org/authors/?q=ai:santiago.regivan-h-nunes"Vieira, Thiago"https://zbmath.org/authors/?q=ai:vieira.thiagoSummary: Recently, the first author et al. [Inf. Sci. 562, 180--199 (2021; Zbl 1528.03221)] generalized the notion of overlap functions in the context of lattices and introduced a weaker definition, called quasi-overlap, that originates from the removal of the continuity condition. In this paper, we introduce the concept of residuated implications related to quasi-overlap functions on lattices and prove some related properties. We also show that the class of quasi-overlap functions that fulfill the residuation principle is the same class of continuous functions according to a Scott topology on lattices. Scott continuity and the notion of densely ordered posets are used to generalize a classification theorem for residuated quasi-overlap functions on lattices. Conjugated quasi-overlaps are also considered.Atoms of the lattices of residuated mappingshttps://zbmath.org/1537.060012024-07-25T18:28:20.333415Z"Kaarli, Kalle"https://zbmath.org/authors/?q=ai:kaarli.kalle"Radeleczki, Sándor"https://zbmath.org/authors/?q=ai:radeleczki.sandorSummary: Given a lattice \(L\), we denote by \(\mathrm{Res}(L)\) the lattice of all residuated maps on \(L\). The main objective of the paper is to study the atoms of \(\mathrm{Res}(L)\) where \(L\) is a complete lattice. Note that the description of dual atoms of \(\mathrm{Res}(L)\) easily follows from earlier results of
\textit{Z. Shmuely} [Pac. J. Math. 54, No. 2, 209--225 (1974; Zbl 0275.06003)]. We first consider lattices \(L\) for which all atoms of \(\mathrm{Res}(L)\) are mappings with 2-element range and give a sufficient condition for this. Extending this result, we characterize these atoms of \(\mathrm{Res}(L)\) which are weakly regular residuated maps in the sense of
\textit{T. S. Blyth} and \textit{M. F. Janowitz} [Residuation theory. Oxford: Pergamon Press; Warszawa: PWN - Polish Scienctific Publishers, (1972; Zbl 0301.06001)]. In the rest of the paper we investigate the atoms of \(\mathrm{Res}(M)\) where \(M\) is the lattice of a finite projective plane, in particular, we describe the atoms of \(\mathrm{Res}(F)\), where \(F\) is the lattice of the Fano plane.Compatibilities between continuous semilatticeshttps://zbmath.org/1537.060022024-07-25T18:28:20.333415Z"Mykytsey, O. Ya."https://zbmath.org/authors/?q=ai:mykytsey.o-ya"Koporkh, K. M."https://zbmath.org/authors/?q=ai:koporkh.k-mSummary: We define compatibilities between continuous semilattices as Scott continuous functions from their pairwise cartesian products to \(\{0,1\}\) that are zero preserving in each variable. It is shown that many specific kinds of mathematical objects can be regarded as compatibilities, among them monotonic predicates, Galois connections, completely distributive lattices, isotone mappings with images being chains, semilattice morphisms etc. Compatibility between compatibilities is also introduced, it is shown that conjugation of non-additive real-valued or lattice valued measures is its particular case.Congruence pairs of decomposable MS-algebrashttps://zbmath.org/1537.060052024-07-25T18:28:20.333415Z"El-Assar, Sanaa"https://zbmath.org/authors/?q=ai:el-assar.sanaa"Badawy, Abd El-Mohsen"https://zbmath.org/authors/?q=ai:badawy.abd-el-mohsenSummary: In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [the second author et al., Math. Slovaca 70, No. 6, 1275--1288 (2020; Zbl 1505.06013)]). They show that every congruence relation \(\theta\) on a decomposable MS-algebra \(L\) can be uniquely determined by a congruence pair \((\theta_1,\theta_2)\), where \(\theta_1\) is a congruence on the de Morgan subalgebra \(L^{\circ\circ}\) of \(L\) and \(\theta_2\) is a lattice congruence on the sublattice \(D(L)\) of \(L\). They obtain certain congruence pairs of a decomposable MS-algebra \(L\) via central elements of \(L\). Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.Polynomial functions over dual numbers of several variableshttps://zbmath.org/1537.130142024-07-25T18:28:20.333415Z"Al-Maktry, Amr Ali Abdulkader"https://zbmath.org/authors/?q=ai:al-maktry.amr-ali-abdulkaderIn a commutative ring \(A\), a polynomial function on \(A\) is a function \([f]_A\): \(a\in A \mapsto f(a)\) where \(f\) belongs to the polynomial ring \(A[x]\). If \([f]_A\) is one-to-one, then \([f]_A\) is called a polynomial permutation and \(f\) is called a permutation polynomial. The set of polynomial functions on \(A\) is denoted by \(\mathcal{F}(A)\) and the set of polynomial permutations on \(A\) is denoted by \(\mathcal{P}(A)\). We also let \(N_A=\{f\in A[x]\; :\; [f]_A=[0]_A\}\) be the set of null polynomials, and we set \(N_A'=\{f\in N_A\; :\; [f']_A=[0]_A\}\), where \(f'\) is the formal derivative of \(f\); both sets are ideals of \(A[x]\). In the following, \(R\) is a finite commutative ring with unity, \(k\) is a positive integer and \(A=R[\alpha_1,\dots,\alpha_k]\) is the ring of dual numbers of \(k\) variables over \(R\) where, for every \(i,j\) in \(\{1,\dots,k\}\), \(\alpha_i\alpha_j=0\). The ring \(R[\alpha_1,\dots,\alpha_k]\) is isomorphic to \(R[x_1,\dots,x_k]/I\), where \(I\) is the ideal generated by the set \(\{x_ix_j\; : \; (i,j)\in\{1,\dots,k\}^2\}\). Every \(f\) in \(A\) will be written as \(f=f_0+f_1\alpha_1+\cdots +f_k\alpha_k\), where \(f_0,f_1,\dots,f_k\) belong to \(R[x]\). First the author proves that \(f\in N_A\) if, and only if, \(f_0\in N_R'\) and \(f_1,\dots, f_k\) lie in \(N_R\). We can't summarize the 11 theorems, 10 propositions, 8 corollaries, 7 lemmas and 7 remarks of this paper, however we give some examples.
-- The cardinal \(|\mathcal{F}(A)|\) of \(\mathcal{F}(A)\) is equal to \([R[x]\! :\!N_R'].[R[x]\! :\!N_R]^k\).
-- Let \(h_1\in N_A\), \(h_2\in N_R\) be monic polynomials and \(d_1=\mbox{deg }h_1\), \(d_2=\mbox{deg }h_2\), then for every \(F\in \mathcal{P}(A)\) there are \(f_0,f_1,\dots,f_k\) in \(R[x]\) with \(\mbox{deg }f_0<d_1,\mbox{deg }f_1<d_2,\dots,\mbox{deg }f_k<d_2\) such that \(F=[f_0+f_1\alpha_1+\cdots +f_k\alpha_k]_A\). Furthermore, if \(h_1\in R[x]\) and there is \(f\in R[x]\) such that \(F=[f]_A\), then \([f]_R=[g]_R\) and \([f']_R=[g']_R\) for some \(g\in R[x]\) with \(\mbox{deg }g<d_1\).
-- The polynomial \(f\) lies in \(\mathcal{P}(A)\) if, and only if, \(f_0\in \mathcal{P}(R)\) and, for every \(a\in R\), \(f_0'(a)\) is a not a zero divisor in \(R\).
-- If \(R\) is a direct sum of local rings which are not fields, then \(f\in \mathcal{P}(A)\Leftrightarrow f_0\in \mathcal{P}(R)\).
-- The stabilizer of \(R\) in \(\mathcal{P}(A)\) is the set \(St(R)=\{F\in \mathcal{P}(A)\; :\; \forall a\in R\; F(a)=a\}=\{F\in\mathcal{P}(A)\; :\; \exists h\in N_R\; F=[x+h]\}\). The set \(St(R)\) is a normal subgroup of \(\mathcal{P}_R(A)=\{[f]_A\; :\; f\in R[x]\}\), (\(\mathcal{P}_R(A)\) is isomorphic to \(\mathcal{P}_R(R[\alpha_i])\), for any \(i\in\{1,\dots,k\}\)). We have \(|\mathcal{P}(A)|=|\mathcal{F}(R)|^k.|\mathcal{P}(R)|.St(R)|\).
-- If \(R=\mathbb{F}_q\), then \(|\mathcal{P}(A)|=q!(q-1)^q q^{kq}\) and \(|St(\mathbb{F}_q)|=(q-1)^q\).
In the last section, the author provides an algorithm which decides whether or not a given function on \(A\) is a polynomial function and if this is so it returns its polynomial representation.
Reviewer: Gerard Leloup (Le Mans)Lattice properties of partial orders for complex matrices via orthogonal projectorshttps://zbmath.org/1537.150092024-07-25T18:28:20.333415Z"Cimadamore, C. R."https://zbmath.org/authors/?q=ai:cimadamore.cecilia-rossana"Rueda, L. A."https://zbmath.org/authors/?q=ai:rueda.laura-a"Sauras-Altuzarra, L."https://zbmath.org/authors/?q=ai:sauras-altuzarra.lorenzo"Thome, N."https://zbmath.org/authors/?q=ai:thome.nestorApart from the well-known partial ordering on the Hermitian matrices there are other partial orderings on various subsets of the ring \(\mathbb{C} ^{n\times n}\) of \(n\times n\) complex matrices. The present authors investigate three previously studied orderings in a unified way. The partial orderings (i.e., the left star order, the star order and the core partial order) are denoted by \(\overset{x}{\leq }\) where \(x\in \{\ell \ast ,\ast ,(\#)\}\) and are defined as follows:
(a) \(A\overset{x}{\leq }B\) for \((x=\ell \ast )\) if and only if \(A^{\ast }A=A^{\ast }B\) and \(A=BB^{\dag }A\) where \(B^{\dag }\) is the Moore-Penrose inverse of \(B\);
(b) \(A\overset{x}{\leq }B\) for \((x=\ast )\) if and only if \(A^{\ast }A=A^{\ast }B\) and \(AA^{\ast }=BA^{\ast }\);
(c) \(A\overset{x}{\leq }B\) for \((x=(\#))\) if and only if \(A^{\ast }A=A^{\ast }B\) and \(BA=A^{2}\).
In (a) and (b) the ordering is defined on the whole of \( \mathbb{C}^{n\times n}\) but in (c) it is only defined on the set of matrices of index at most \(1\), that is, when \(\mathrm{rank}(C^{2})=\mathrm{rank}(C)\).
All three partial orderings are preserved under unitary similarity and it is known (but not obvious) that each of the partial orderings is a lattice. The aim of the present paper is to describe the down-sets, namely the sublattices of the form \[[0,B]^{x}:=\left\{ A~|~0 \overset{x}{\leq }A\overset{x}{\leq }B\right\} \] for arbitrary \(B\). By using the \textit{R. E. Hartwig} and \textit{K. Spindelboeck}'s decomposition of \(B\) [Linear Multilinear Algebra 14, 241--256 (1983; Zbl 0525.15006)], the authors provide simpler proofs of previously proved theorems as well as some new results. For example, they show that \([0,B]^{\ast }\) is a sublattice of \([0,B]^{\ell \ast }\) for all \(B\) (Theorem 3.7) and that \([0,B]^{\ast }\) is an orthomodular lattice of finite height which is distributive if and only if it is a Boolean algebra (Proposition 3.8). Similarly, if \(B\) is of index at most 1, then \( [0,B]^{(\#)}\) is a sublattice of \([0,B]^{\ell \ast }\) (Theorem 3.14).
Reviewer: John D. Dixon (Ottawa)Deformations and extensions of BiHom-alternative algebrashttps://zbmath.org/1537.170542024-07-25T18:28:20.333415Z"Chtioui, Taoufik"https://zbmath.org/authors/?q=ai:chtioui.taoufik"Mabrouk, Sami"https://zbmath.org/authors/?q=ai:mabrouk.sami"Makhlouf, Abdenacer"https://zbmath.org/authors/?q=ai:makhlouf.abdenacerSummary: The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and \(T_\theta\)-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutationshttps://zbmath.org/1537.200932024-07-25T18:28:20.333415Z"Yu, Houyi"https://zbmath.org/authors/?q=ai:yu.houyiThe (left) weak order is a useful tool in the combinatorial study of Coxeter groups. It can be defined as the suffix order of reduced expressions of the group elements, or more combinatorially as the inclusion order of the right associated reflection sets of the group elements. The general structure of the weak order on an arbitrary Coxeter group was first systematically investigated by \textit{A. Björner} [Contemp. Math. 34, 175--195 (1984; Zbl 0594.20029)], and extensively studied in in several subsequent works.
In the paper under review, the author gives a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a larger hyperoctahedral group. As an application, the author investigates the monomial basis for the Hopf algebra \(\mathfrak{H}\mathrm{Sym}\) of signed permutations, related to the fundamental basis via Möbius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a monomial basis element under the descent map from \(\mathfrak{H}\mathrm{Sym}\) to the algebra of type \(\mathsf{B}\) quasi-symmetric functions is either zero or a monomial quasi-symmetric function of type \(\mathsf{B}\).
Reviewer: Egle Bettio (Venezia)