scholarly journals On the Möbius function of the locally finite poset associated with a numerical semigroup

2012 ◽  
Vol 87 (2) ◽  
pp. 313-330 ◽  
Author(s):  
Jonathan Chappelon ◽  
Jorge Luis Ramírez Alfonsín
Keyword(s):  
Numerical Semigroup ◽  
Möbius Function ◽  
Locally Finite ◽  
Finite Poset ◽  
Mobius Function ◽  
Locally Finite Poset

scholarly journals The Möbius function of generalized subword order

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Peter R. W. McNamara ◽  
Bruce E. Sagan
Keyword(s):  
Partial Order ◽  
Finite Length ◽  
Simple Formula ◽  
Möbius Function ◽  
Finite Poset ◽  
Mobius Function ◽  
Nous Permet ◽  
International Audience ◽  
Uniform Manner ◽  
A Chain

International audience Let $P$ be a poset and let $P^*$ be the set of all finite length words over $P$. Generalized subword order is the partial order on $P^*$ obtained by letting $u≤ w$ if and only if there is a subword $u'$ of $w$ having the same length as $u$ such that each element of $u$ is less than or equal to the corresponding element of $u'$ in the partial order on $P$. Classical subword order arises when $P$ is an antichain, while letting $P$ be a chain gives an order on compositions. For any finite poset $P$, we give a simple formula for the Möbius function of $P^*$ in terms of the Möbius function of $P$. This permits us to rederive in an easy and uniform manner previous results of Björner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in $P^*$ for any finite $P$ of rank at most 1. Soit $P$ un ensemble partiellement ordonné et soit $P^*$ l'ensemble des mots de longueur finie sur $P$. On définit l'ordre des sous-mots généralisé comme l'ordre partiel sur $P^*$ obtenu en posant $u≤ w$ s'il existe un sous-mot $u'$ de $w$ ayant la même longueur que $u$, tel que chaque élément de $u$ soit plus petit ou égal à l'élément correspondant de $u'$ dans l'ordre partiel sur $P$. L'ordre des sous-mots classique correspond au cas où $P$ est une antichaîne ; tandis que si P est une chaîne, on obtient un ordre sur les compositions. Pour tout ensemble partiellement ordonné fini $P$, nous donnons une formule simple pour la fonction de Möbius de $P^*$ en fonction de celle de $P$. Cela nous permet de retrouver de manière simple et uniforme des résultats de Björner, Sagan et Vatter, et de Tomie. Nous sommes aussi en mesure de déterminer le type d'homotopie de tous les intervalles de $P^*$ pour n'importe quel $P$ fini de rang au plus 1.

scholarly journals On incidence algebras and directed graphs

2002 ◽  
Vol 31 (5) ◽  
pp. 301-305
Author(s):  
Ancykutty Joseph
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Locally Finite ◽  
Incidence Algebra ◽  
Finite Poset ◽  
Principal Ideals ◽  
Incidence Algebras ◽  
Directed Paths ◽  
Multiple Edges ◽  
Locally Finite Poset

The incidence algebraI(X,ℝ)of a locally finite poset(X,≤)has been defined and studied by Spiegel and O'Donnell (1997). A poset(V,≤)has a directed graph(Gv,≤)representing it. Conversely, any directed graphGwithout any cycle, multiple edges, and loops is represented by a partially ordered setVG. So in this paper, we define an incidence algebraI(G,ℤ)for(G,≤)overℤ, the ring of integers, byI(G,ℤ)={fi,fi*:V×V→ℤ}wherefi(u,v)denotes the number of directed paths of lengthifromutovandfi*(u,v)=−fi(u,v). WhenGis finite of ordern,I(G,ℤ)is isomorphic to a subring ofMn(ℤ). Principal idealsIvof(V,≤)induce the subdigraphs〈Iv〉which are the principal idealsℐvof(Gv,≤). They generate the idealsI(ℐv,ℤ)ofI(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.

Reduced finitary incidence algebras and their automorphisms

2021 ◽  
pp. 1-30
Author(s):  
Manfred Dugas ◽  
Daniel Herden ◽  
Jack Rebrovich
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Locally Finite ◽  
Incidence Algebra ◽  
Finite Poset ◽  
Incidence Algebras ◽  
Special Cases ◽  
Locally Finite Poset

Let [Formula: see text] denote the incidence algebra of a locally finite poset [Formula: see text] over a field [Formula: see text] and [Formula: see text] some equivalence relation on the set of generators of [Formula: see text]. Then [Formula: see text] is the subset of [Formula: see text] of all the elements that are constant on the equivalence classes of [Formula: see text]. If [Formula: see text] satisfies certain conditions, then [Formula: see text] is a subalgebra of [Formula: see text] called a reduced incidence algebra. We extend this notion to finitary incidence algebras [Formula: see text] for any poset [Formula: see text]. We investigate reduced finitary incidence algebras [Formula: see text] and determine their automorphisms in some special cases.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

scholarly journals From explicit estimates for primes to explicit estimates for the Möbius function

2013 ◽  
Vol 157 (4) ◽  
pp. 365-379 ◽  
Author(s):  
Olivier Ramaré
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

scholarly journals A natural proof of the cyclotomic identity

1990 ◽  
Vol 42 (2) ◽  
pp. 185-189 ◽  
Author(s):  
D.E. Taylor
Keyword(s):  
Natural Isomorphism ◽  
Möbius Function ◽  
Mobius Function ◽  
Image Position

The cyclotomic identitywhere and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Sign in / Sign up

Export Citation Format

Share Document