Recent zbMATH articles in MSC 06Ahttps://www.zbmath.org/atom/cc/06A2021-04-16T16:22:00+00:00WerkzeugAssociative, idempotent, symmetric, and order-preserving operations on chains.https://www.zbmath.org/1456.060042021-04-16T16:22:00+00:00"Devillet, Jimmy"https://www.zbmath.org/authors/?q=ai:devillet.jimmy"Teheux, Bruno"https://www.zbmath.org/authors/?q=ai:teheux.brunoAssociativity of binary operations is important because numerous algebraic structures are defined with associative operations as semigroups, groups, rings, lattices, etc. Associativity has been considered in conjunction with other properties such as idempotency or quasitriviality. The authors characterize the associative, idempotent, symmetric, and order-preserving binary operations on finite chains in terms of their associated semilattice order. They prove that the number of associative, idempotent, symmetric and order-preserving operations on an \(n\)-element chain is just the \(n\)-th Catalan number.
Reviewer: Ivan Chajda (Přerov)On binary relations induced from overlap and grouping functions.https://www.zbmath.org/1456.030822021-04-16T16:22:00+00:00"Qiao, Junsheng"https://www.zbmath.org/authors/?q=ai:qiao.junshengSummary: In this paper, firstly, we introduce the binary relation \(\preceq_O\) derived from an overlap function \(O\). And then, we show that \(\preceq_O\) does not satisfy the reflexivity, anti-symmetry and transitivity naturally, and investigate the conditions under which \(\preceq_O\) can become a reflexive, anti-symmetric or transitive relation, respectively. In particular, we obtain a necessary and sufficient condition for the binary relation \(\preceq_O\) becoming a partial order on the unit interval \([0, 1]\). Finally, we give an analogous discussion for the binary relations induced from grouping functions.A new characterization of \(\mathcal{V} \)-posets.https://www.zbmath.org/1456.060012021-04-16T16:22:00+00:00"Cooper, Joshua"https://www.zbmath.org/authors/?q=ai:cooper.joshua-n"Gartland, Peter"https://www.zbmath.org/authors/?q=ai:gartland.peter"Whitlatch, Hays"https://www.zbmath.org/authors/?q=ai:whitlatch.haysThe paper defines a finite poset \(P\) to be ``autonomous''
if there exists a directed acyclic graph \(D\) with
adjacency matrix \(U\) whose transitive closure is \(P\),
with the property that any total ordering of the vertices of
\(D\) so that Gaussian elimination of \(U^tU\)
proceeds without row swaps is a linear extension of \(P\).
The main theorem of the paper is that a finite poset is autonomous
if and only if it is induced \(N\)-free and induced bowtie-free.
This class of posets has a ``series-parallel''
type of characterization:
by a theorem of \textit{T. Hasebe} and \textit{S. Tsujie} [J. Algebr. Comb. 46, No. 3--4, 499--515 (2017; Zbl 1423.06011)] the
class of finite induced \(N\)-free and induced bowtie-free
posets is the smallest isomorphism-closed class of posets
which (i) contains the \(1\)-element poset,
(ii) is closed under the formation of disjoint union, and
(iii) is closed under adjoining new least or largest elements.
Reviewer: Keith Kearnes (Boulder)Lie triple derivations of incidence algebras.https://www.zbmath.org/1456.160402021-04-16T16:22:00+00:00"Wang, Danni"https://www.zbmath.org/authors/?q=ai:wang.danni"Xiao, Zhankui"https://www.zbmath.org/authors/?q=ai:xiao.zhankuiLet \(A\) be an associative algebra over \(\mathcal{R}\), a commutative ring with unit, and let \(Z(A)\) denote the center of \(A\). An \(\mathcal{R}\)-linear map \(L\colon A\rightarrow A\) is called a Lie triple derivation if \(L([[x,y],z])=[[L(x),y],z]+[[x,L(y)],z]+[[x,y],L(z)]\) for all \(x,y,z\in A\) and where \([x,y]\) denotes the commutator of \(x\), \(y\). A Lie triple derivation is proper if it is of the form \(D+F\), where \(D\colon A\rightarrow A\) is a derivation, and \(F\colon A\rightarrow Z(A)\) is an \(\mathcal{R}\)-linear map.
From now on, \(\mathcal{R}\) denotes a \(2\)-torsion free commutative ring with unit, \(X\) denotes a locally finite preordered set, and \(I(X,\mathcal{R})\) denotes the incidence algebra of \(X\) over \(\mathcal{R}\). In the paper under review, the authors prove that if \(X\) consists of a finite number of connected components, then every Lie triple derivation of \(I(X,\mathcal{R})\) is proper.
Reviewer: Małgorzata E. Hryniewicka (Białystok)Birational rowmotion and Coxeter-motion on minuscule posets.https://www.zbmath.org/1456.051792021-04-16T16:22:00+00:00"Okada, Soichi"https://www.zbmath.org/authors/?q=ai:okada.soichiSummary: Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistics. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.Sign variation and descents.https://www.zbmath.org/1456.051842021-04-16T16:22:00+00:00"Bergeron, Nantel"https://www.zbmath.org/authors/?q=ai:bergeron.nantel"Dermenjian, Aram"https://www.zbmath.org/authors/?q=ai:dermenjian.aram"Machacek, John"https://www.zbmath.org/authors/?q=ai:machacek.john-mSummary: For any \(n > 0\) and \(0 \leqslant m < n\), let \(P_{n,m}\) be the poset of projective equivalence classes of \(\{-,0,+\}\)-vectors of length \(n\) with sign variation bounded by \(m\), ordered by reverse inclusion of the positions of zeros. Let \(\Delta_{n,m}\) be the order complex of \(P_{n,m}\). A previous result from the third author [``Boundary measurement and sign variation in real projective space'', Preprint, \url{arXiv:1909.04640}] shows that \(\Delta_{n,m}\) is Cohen-Macaulay over \(\mathbb{Q}\) whenever \(m\) is even or \(m = n-1\). Hence, it follows that the \(h\)-vector of \(\Delta_{n,m}\) consists of nonnegative entries. Our main result states that \(\Delta_{n,m}\) is partitionable and we give an interpretation of the \(h\)-vector when \(m\) is even or \(m = n-1\). When \(m = n-1\) the entries of the \(h\)-vector turn out to be the new Eulerian numbers of type \(D\) studied by \textit{A. Borowiec} and \textit{W. Młotkowski} [Electron. J. Comb. 23, No. 1, Research Paper P1.38, 13 p. (2016; Zbl 1382.05004)]. We then combine our main result with Klee's generalized Dehn-Sommerville relations to give a geometric proof of some facts about these Eulerian numbers of type \(D\).Classifications of \(\Gamma\)-colored \(d\)-complete posets and upper \(P\)-minuscule Borel representations.https://www.zbmath.org/1456.051762021-04-16T16:22:00+00:00"Strayer, Michael C."https://www.zbmath.org/authors/?q=ai:strayer.michael-cSummary: The \(\Gamma\)-colored \(d\)-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify \(\Gamma\)-colored \(d\)-complete posets which specifies the structure of the associated representations. We show that finite \(\Gamma\)-colored \(d\)-complete posets are precisely the dominant minuscule heaps of \textit{J. R. Stembridge} [J. Algebra 235, No. 2, 722--743 (2001; Zbl 0973.17034)]. These heaps are reformulations and extensions of the colored \(d\)-complete posets of \textit{R. A. Proctor} [J. Algebra 213, No. 1, 272--303 (1999; Zbl 0969.05068)]. We also show that connected infinite \(\Gamma\)-colored \(d\)-complete posets are precisely order filters of the connected full heaps of \textit{R. M. Green} [Combinatorics of minuscule representations. Cambridge: Cambridge University Press (2013; Zbl 1320.17005)].Towards an algebra for cascade effects.https://www.zbmath.org/1456.060072021-04-16T16:22:00+00:00"Adam, Elie M."https://www.zbmath.org/authors/?q=ai:adam.elie-m"Dahleh, Munther A."https://www.zbmath.org/authors/?q=ai:dahleh.munther-a"Ozdaglar, Asuman"https://www.zbmath.org/authors/?q=ai:ozdaglar.asuman-eSummary: We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied \textit{models} of cascade effects.
We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express \textit{global} systems through combinations of \textit{local} systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of \(\mu\)-rank to capture the energy of a system, and understand the minimal amount of \textit{effort} required to fail a system, termed \textit{resilience}. We deduce a dual notion of \textit{fragility} and show that the combination of systems sets a limit on the amount of fragility inherited.Two applications of a generalization of an asymptotic fixed point theorem.https://www.zbmath.org/1456.060022021-04-16T16:22:00+00:00"Herzog, Gerd"https://www.zbmath.org/authors/?q=ai:herzog.gerd"Kunstmann, Peer Chr."https://www.zbmath.org/authors/?q=ai:kunstmann.peer-christianSummary: We present a variant of an asymptotic version of the Abian-Brown Fixed Point Theorem, and applications to recursively defined sequences and Hammerstein integral equations.Poset loops.https://www.zbmath.org/1456.060032021-04-16T16:22:00+00:00"Smith, J. D. H."https://www.zbmath.org/authors/?q=ai:smith.jonathan-d-hSummary: Given a ring and a locally finite poset, an \textit{incidence loop} or \textit{poset loop} is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.Quantified conjunctive queries on partially ordered sets.https://www.zbmath.org/1456.680962021-04-16T16:22:00+00:00"Bova, Simone"https://www.zbmath.org/authors/?q=ai:bova.simone"Ganian, Robert"https://www.zbmath.org/authors/?q=ai:ganian.robert"Szeider, Stefan"https://www.zbmath.org/authors/?q=ai:szeider.stefanSummary: We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already NP-hard on a certain fixed poset, and investigate structural properties of posets yielding fixed-parameter tractability when the problem is parameterized by the query. Our main algorithmic result is that model checking quantified conjunctive queries on posets of bounded width is fixed-parameter tractable (the width of a poset is the maximum size of a subset of pairwise incomparable elements). We complement our algorithmic result by complexity results with respect to classes of finite posets in a hierarchy of natural poset invariants, establishing its tightness in this sense.
For the entire collection see [Zbl 1318.68014].Higher derivations of finitary incidence algebras.https://www.zbmath.org/1456.160262021-04-16T16:22:00+00:00"Kaygorodov, Ivan"https://www.zbmath.org/authors/?q=ai:kaigorodov.i-b"Khrypchenko, Mykola"https://www.zbmath.org/authors/?q=ai:khrypchenko.mykola-s"Wei, Feng"https://www.zbmath.org/authors/?q=ai:wei.fengA sequence of additive maps \((d_n)_{n\in \mathbb N}\) on a unital ring \(R\) is called a higher derivation if the identities
\[ d_0(x)=x\quad \hbox{ and } \quad d_n(xy) = \sum_{k=0}^n d_k(x)d_{n-k}(y)\] hold. Examples include the sequence of additive maps \(d_n\colon x\mapsto r^{n-1}(rx-xr)\), with an element \(r\in R\) kept fixed, as well as, when \(R\) is an algebra over a field with characteristic \(0\), the sequence \((\frac{1}{n!}d^n)_{n\in\mathbb N}\) with \(d\colon R\to R\) being a usual derivation on \(R\).
In fact, higher derivations are in one-to-one, onto correspondence with those automorphisms \(\alpha\) of the ring of formal power series \(R[[t]]\) which fix an indeterminate \(t\) and map each \(x\in R\subseteq R[[t]]\) into the set \(x+t R[[t]]\); the correspondence is given by \(\alpha(x)=\sum d_{n}(x)t^n\); \(x\in R\subseteq R[[t]]\).
The main result of the paper under review describes the form of \(R\)-linear higher derivations on finitary incidence algebras \(FI(R)\) over commutative unital rings \(R\). Here, by definition, \(FI(R)\) is an \(R\)-algebra of \(R\)-valued functions with domain consisting of all pairs \((x,y)\), ordered in a given preordered set \(P\), which have a finite support when restricted to each of the subsets \(\Omega_{(x,y)}:=\{(u,v)\in P^2;\;\; x\le u<v\le y\}\). The \(R\)-module structure on \(FI(R)\) is standard and the multiplication is convolution-like
\[(f\ast g)(x,y):=\sum_{x\le z\le y} f(x,z)g(z,y).\]
Reviewer: Bojan Kuzma (Ljubljana)Lusternik-Schnirelmann category of relation matrices on finite spaces and simplicial complexes.https://www.zbmath.org/1456.550032021-04-16T16:22:00+00:00"Tanaka, Kohei"https://www.zbmath.org/authors/?q=ai:tanaka.koheiLet \(\phi\colon K\to L\) be a simplicial map between finite simplicial complexes. The simplicial LS category of \(f\), denoted \(\mathrm{scat}(f)\) is the smallest non-negative integer \(n\) such that there exists a cover \(\{U_i\}_{i=0}^n\) of \(X\) where \(f|U_i\) is contiguous to the constant map for every \(0\leq i\leq n\). This is a simplicial analogue to the classic Lusternik-Schnirelmann category, defined in a similar way, of a continuous map between topological spaces. A recent result of \textit{J. González} [New York J. Math. 24, 279--292 (2018; Zbl 1394.55004)] shows that if \(\lambda_k\colon \mathrm{sd}^k(K)\to K\) is the simplicial approximation from the \(k^{th}\) barycentric subdivision of \(K\), then \(\mathrm{scat}(f\circ \lambda_K)=\mathrm{cat}(|f|)\) for sufficiently large \(k\). In other words, \(\mathrm{scat}(f)\) can be thought of as approximating \(\mathrm{cat}(|f|)\).
Using the result of González as well as some of the work of \textit{G. Raptis} [Homology Homotopy Appl. 12, No. 2, 211--230 (2010; Zbl 1215.18017)] on the homotopy theory of posets, the author of the paper under review develops a combinatorial method to approximate the LS category of a continuous map \(\phi\colon |K|\to |L|\) in practice. This is accomplished by studying the principal relational matrix of a map between finite spaces associated to the complexes. Once this matrix is obtained, one may perform ``operations'' on it, akin to elementary row operations, in order to reduce the matrix and compute the simplicial category. Some computations are given and it is pointed out that although in theory this method is promising since the number of open sets is theoretically finite, this may be difficult in practice since the barycentric subdivision grows factorially in the number of simplices.
Reviewer: Nicholas A. Scoville (Collegeville)Computing character degrees via a Galois connection.https://www.zbmath.org/1456.200052021-04-16T16:22:00+00:00"Lewis, Mark L."https://www.zbmath.org/authors/?q=ai:lewis.mark-l"McVey, John K."https://www.zbmath.org/authors/?q=ai:mcvey.john-kSummary: In a previous paper, the second author established that, given finite fields \(F < E\) and certain subgroups \(C\leq E^\times\), there is a Galois connection between the intermediate field lattice \(\{L \mid F \leq L \leq E\}\) and \(C\)'s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product \(C\rtimes\mathrm{Gal}(E/F)\). However, the analysis when \(|F|\) is a Mersenne prime is more complicated, so certain cases were omitted from that paper. The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group \(C\rtimes \mathrm{Gal}(E/F)\), we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup \(C\leq E^\times\) which satisfies the condition that every prime dividing \(|E^\times :C|\) divides \(|F^\times|\).