Maximal Chains Research Articles

Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Let ℜ be the finite commutative ring with unity and <i>I<sub>s</sub></i> be the S-prime ideal of a ring ℜ. The set <i>L<sub>s</sub></i> forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (<i>L<sub>s</sub></i>, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.

Combinatorial Results on Barcode Lattices

AbstractA barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

On the Shelter of a Poset & Its Algebraic Applications

For a poset \(P = C_a \times C_b\), a subset \(A \subseteq P\) is called a chain blocker for \(P\) if \(A\) is inclusion-wise minimal with the property that every maximal chain in \(P\) contains at least one element of \(A\), where \(C_i\) is the chain \(1 < \cdots < i\). In this article, we define the shelter of the poset \(P\) to give a complete description of all chain blockers of \(C_5 \times C_b\) for \(b \geq 1\).

Brain chains as topological signatures for Alzheimer’s disease

AbstractTopology is providing new insights for neuroscience. For instance, graphs, simplicial complexes, directed graphs, flag complexes, persistent homology and convex covers have been used to study functional brain networks, synaptic connectivity, and hippocampal place cell codes. We propose a topological framework to study the evolution of Alzheimer’s disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study these filtrations we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall’s $$d_K$$ d K metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer’s disease.

Transposed Poisson structures on Lie incidence algebras

Let X be a finite connected poset, K a field of characteristic zero and I(X,K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 12-derivation of I(X,K) decomposes into the sum of a central-valued 12-derivation, an inner 12-derivation and a 12-derivation associated with a map σ:X<2→K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I(X,K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ:Xe2→K, where Xe2 is the set of (x,y)∈X2 such that x<y is a maximal chain not contained in a cycle.

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Maximal chain-based Choquet-like integrals

The original idea of the discrete Choquet integral on nonnegative reals is based on a distinguished permutation making the inputs reordering nonincreasingly or nondecreasingly. This permutation generates the weights from the given capacity, and then a weighted arithmetic mean yields the value of the corresponding Choquet integral. Since the permutation relates to a maximal chain, we introduce a concept of MCC-integrals using the setting of maximal chains on the power set of a finite set. This approach provides a unified framework for several integrals of nonnegative vectors with respect to games. We study basic properties and various representations of these MCC-integrals subsuming known results for the Choquet integral as a special case. For supermodular games it is shown that the Choquet integral is the minimum of all product-based MCC-integrals. Further, we discuss symmetric and asymmetric extensions for real-valued inputs providing new insight into the construction of fuzzy integral quadruplets. During the way, we point out certain inaccuracies in the literature.

Eventual Fitting Shadowing Property for Hyperbolic Dynamical Systems

Let be a diffeomorphism map on a closed smooth manifold for dimension . We explain in this work any chain transitive set of generic diffeomorphism , if a diffeomorphism has another type of shadowing property is called, the eventual shadowing property on locally maximal chain transitive set, then is hyperbolic. In general, the eventual fitting shadowing property is not fulfilled in hyperbolic dynamical systems (satisfy in case L is Anosov diffeomorphism map) . In this paper, several concepts were presented. These concepts can be re-examined on other important spaces, and their impact on finding dynamical characteristics that may be employed in solving some mathematical problems.

Supersolvable saturated matroids and chordal graphs

A matroid is supersolvable if it has a maximal chain of flats, each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroidal context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of tree-width.

From trees to barcodes and back again II: Combinatorial and probabilistic aspects of a topological inverse problem

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on n+1 leaf nodes fall into (2n−1)!! distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2−n. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

Combinatorics Arising from Lax Colimits of Posets

In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the 2-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.

A new algorithm to compute fuzzy subgroups of a finite group

&lt;abstract&gt;&lt;p&gt;The enumeration of fuzzy subgroups is a complex problem. Several authors have computed results for special instances of groups. In this paper, we present a novel algorithm that is designed to enumerate the fuzzy subgroups of a finite group. This is achieved through the computation of maximal chains of subgroups. This approach is also useful for writing a program to compute the number of fuzzy subgroups. We provide further elucidation by computing the number of fuzzy subgroups of the groups $ Q_8 $ and $ D_8 $.&lt;/p&gt;&lt;/abstract&gt;

Counting weighted maximal chains in the circular Bruhat order

The totally nonnegative Grassmannian Gr(k,n)≥0 is the subset of the real Grassmannian Gr(k,n) consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of Postnikov's positroid cell decomposition of Gr(k,n)≥0[10]. We provide a closed formula for the sum of its weighted chains in the spirit of Stembridge [11].

A new class of decomposition integrals on finite spaces

A new type of decomposition integral is introduced by using a family of decomposition integrals based on the collections relating to partitions and maximal chains of sets. This new integral extends the Lebesgue integral, and it is different from those well-known decomposition integrals, such as the Choquet, concave, pan-, Shilkret integrals and PC-integral. In the structure of a lattice on the class of decomposition integrals, the introduced decomposition integral is between the Choquet integral and the concave integral, and also between the pan-integral and the concave integral, and it is a lower bound of PC-integral. The coincidences among several well-known integrals and this new integral are also shown.

Maximal Chains in Bond Lattices

Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $G$.&#x0D; In this article, we consider graphs $G$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions.

The Jacobson radical for an inconsistency predicate

As a form of the Axiom of Choice about relatively simple structures (posets), Hausdorff’s Maximal Chain Principle appears to be little amenable to computational interpretation. This received view, however, requires revision: maximal chains are more reminiscent of maximal ideals than it seems at first glance. The latter live in richer algebraic structures (rings), and thus are readier to be put under computational scrutiny. Exploiting this, and of course the analogy between maximal chains and maximal ideals, the concept of Jacobson radical carries over from a ring to an arbitrary set with an abstract inconsistency predicate: that is, a distinguished monotone family of finite subsets. All this makes possible not only to generalise Hausdorff’s principle, but also to express it as a syntactical conservation theorem. The latter, which encompasses the desired computational core of Hausdorff’s principle, is obtained by a generalised inductive definition. The over-all setting is constructive set theory.

The Tamari block lattice: An order on saturated chains in the Tamari lattice

In this article we define a partially-ordered set on equivalence classes of maximal chains of an interval of the Tamari lattice, which we call the Tamari Block Poset. We prove this is a graded lattice. Along the way we discuss some useful characterizations of its elements. In conclusion we define explicit maps between our work and the Higher Stasheff-Tamari orders in dimensions two and three.

On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences

This paper solves the issues of determining the number Fn of fuzzy subsets of a nonempty finite set X. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of X. We derive two closed explicit formulas for Fn, which is the sum of a finite series in the product of binomial numbers or the sum of k-level fuzzy subsets Fn,k by introducing a classification technique. Moreover, these explicit formulas enable us to find the number of the maximal chains of crisp subsets of X. Further, this paper presents some elementary properties of Fn,k and Fn.

Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the W-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

Proper Lie automorphisms of incidence algebras

AbstractLet X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.