相关论文: On the M\"obius function of the locally finite pos…
In this paper, we investigate the M{\"o}bius function $\mu\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study…
The paper presents some results for reducing the computation of the M\"obius functon of a M\"obius category that arises from a combinatorial inverse semigroup to that of locally finite partially ordered sets. We illustrate the computation…
This paper analyzes the M\"obius ($\mu(i)$) function defined on the partially ordered set of triangular numbers ($\mathcal T(i)$) under the divisibility relation. We make conjectures on the asymptotic behavior of the classical M\"obius and…
This paper studies the M\"obius function and related questions about the finiteness of the poset of submodules of semisimple and general modules. We show how to calculate the M\"obius function for semisimple modules based on endomorphism…
Let $B$ be a finite Boolean algebra. Let $\mathcal A$ be the partial order of all implication sublattices of $B$. We will compute the M\"obius function on $\mathcal A$ in two different ways.
We determine the M\"obius function of the poset of compositions of an integer. In fact we give two proofs of this formula, one using an involution and one involving discrete Morse theory. The composition poset turns out to be intimately…
We compute the M\"obius function for the subgroup lattice of the simple Suzuki group $Sz(q)$; this is used to enumerate normal subgroups of certain Hecke groups with quotients isomorphic to $Sz(q)$.
A sequential dynamical system (SDS) consists of a graph, a set of local functions and an update schedule. A linear sequential dynamical system is an SDS whose local functions are linear. In this paper, we derive an explicit closed formula…
We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $\sigma \leq \tau$ whenever the permutation $\tau$ contains an occurrence of the permutation $\sigma$ in…
This is an introduction to the M\"obius function of a poset. The chief novelty is in the exposition. We show how order-preserving maps from one poset to another can be used to relate their M\"obius functions. We derive the basic results on…
We investigate Sarnak's conjecture on the M\"obius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.
We consider the problem of fast zeta and M\"obius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and M\"obius transforms can be computed in $O(e)$ elementary…
We study filters in the partition lattice formed by restricting to partitions by type. The M\"obius function is determined in terms of the easier-to-compute descent set statistics on permutations and the M\"obius function of filters in the…
We study the poset $\mathcal{G}$ of all unlabelled graphs, up to isomorphism, with $H\le G$ if $H$ occurs as an induced subgraph in $G$. We present some general results on the M\"obius function of intervals of $\mathcal{G}$ and some results…
In this paper, we will study the M\"obius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the M\"obius polynomial of the direct product of posets in terms of the…
Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real M\"obius transformations. We define a semigroup $S$ of M\"obius…
The M\"{o}bius function of the subgroup lettice of a finite group $G$ has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let $A$ be a subgroup of the automorphism…
A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M\"obius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a M\"obius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily…
We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical…
We provide a new way to represent numerical semigroups by showing that the position of every Ap\'ery set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this…