A class of matroids derived from saturated chain partitions of partially ordered sets

AbstractThe simultaneously k- and (k − 1)-saturated chain partitions of a finite partially ordered set P determine a matroid Gk(P). This matroid is a gammoid. The identity on P induces a strong map from Gk(P) to Gk + 1(P). This strong map has a linear representation

On the order sequence of a group

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. Abelian groups are uniquely determined by their order sequences, and the poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a group if and only if $|G|$ and $|H|$ are coprime. The paper concludes with a number of open problems.Comment: 22 pages, Comments are most welcom

Signed ring families and signed posets

The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections

Products in the category of forests and p-morphisms via Delannoy paths on Cartesian products

In [2], the authors introduce a technique to compute finite coproducts of finite G\uf6del algebras, i.e. Heyting algebras satisfying the prelinearity axiom (\u3b1 \u2192 \u3b2) 28(\u3b2 \u2192 \u3b1). To do so, they investigate the product in the category opposite to finite G\uf6del algebras: the category of forests and open order-preserving maps, alias p-morphisms, which we denote by F. (A forest is a partially ordered set F such that, for every x in F, the set of lower bounds of x forms a chain, when endowed with the order inherited from F.) To achieve their result, the authors make use of ordered partitions of finite sets and of a specific operation \u2014 called merged-shuffle \u2014 on ordered partitions. In [1, Section 4.2], the authors present an alternative, recursive construction of finite products in the category of forests and open order-preserving maps. In the present work we introduce a further construction of the same finite products, based on products of posets along with a generalization of the combinatorial notion of Delannoy path. The new and most interesting aspect of our construction is that, dually, it uncovers a key relationship between the coproducts of finite G\uf6del algebras and the coproducts in the category of finite distributive lattices. Our main result explains the former coproducts in terms of a construction on the latter; the construction itself is currently best understood via duality using a generalisation of the Delannoy paths. 1. Stefano Aguzzoli, Simone Bova, and Brunella Gerla. Chapter IX: Free algebras and functional representation for fuzzy logics. In Handbook of mathematical fuzzy logic. Volume 2, volume 38 of Stud. Log. (Lond.), pages 713\u2013791. Coll. Publ., London, 2011. 2. Ottavio M. D\u2019Antona and Vincenzo Marra. Computing coproducts of finitely presented G\uf6del algebras. Ann. Pure Appl. Logic, 142(1-3):202\u2013211, 2006

Equality of bond percolation critical exponents for pairs of dual lattices

For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is generalized to a class of lattices that allows the equality of bond percolation critical exponents for lattice-dual pairs to be concluded without performing the computations. The proof uses the substitution method, which involves stochastic ordering of probability measures on partially ordered sets. As a consequence, there is an infinite collection of infinite sets of two-dimensional lattices, such that all lattices in a set have the same critical exponents.Comment: 10 pages, 7 figure