A note on Faigle and Kern’s dual greedy polyhedra

U. Faigle and W. Kern have recently extended the work of their earlier paper and of M. Queyranne, F. Spieksma and F. Tardella and have shown that a dual greedy algorithm works for a system of linear inequalities with {:0,1}-coefficients defined in terms of antichains of an underlying poset and a submodular function on the set of ideals of the poset under some additional condition on the submodular function.?In this note we show that Faigle and Kern’s dual greedy polyhedra belong to a class of submodular flow polyhedra, i.e., Faigle and Kern’s problem is a special case of the submodular flow problem that can easily be solved by their greedy algorithm. Received: February 1999 / Accepted: December 1999?Published online February 23, 2000     

Noncommutative Enumeration in Graded Posets

We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding to odd jumps in flags. In this context, the coefficients of the cd-index provide a graded basis.     

The category of supercontinuous posets

In this paper, we study the order structure—supercontinuous poset, a generalization of completely distributive lattice. The Cartesian product of supercontinuous posets and some other properties of supercontinuous posets are investigated. Also, the case of superalgebraic posets are studied and some remarks on the category of supercontinuous posets are given.     

On the Cofinality of Infinite Partially Ordered Sets: Factoring a Poset into Lean Essential Subsets

We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable cofinality remains open.     

Orthogonal partitions in designed experiments

A survey is given of the statistical theory of orthogonal partitions on a finite set. Orthogonality, closure under suprema, and one trivial partition give an orthogonal decomposition of the corresponding vector space into subspaces indexed by the partitions. These conditions plus uniformity, closure under infima and the other trivial partition give association schemes. Examples covered by the theory include Latin squares, orthogonal arrays, semilattices of subgroups, and partitions defined by the ancestral subsets of a partially ordered set (the poset block structures).Isomorphism, equivalence and duality are discussed, and the automorphism groups given in some cases. Finally, the ideas are illustrated by some examples of real experiments.Dedicated to Hanfried Lenz on the occasion of his 80th birthday.     

States on Orthomodular Amalgamations Over Trees

An amalgamation of bounded involution posets over a strictly directed graph is introduced and states on this amalgamation are studied. We introduce conditions under which the amalgamation induces a structure that is of the same type as that of the amalgamated structures. We also study circumstances under which common properties of the state spaces (such as unital, full, and strongly order determining) of the amalgamated structures are inherited by the amalgamation.In memory of Günter Bruns.     

A short proof of a theorem of Birkhoff

This paper gives a new proof of a theorem of G. Birkhoff: Every group can be represented as the automorphism group of a distributive lattice D; if is finite, D can be chosen to be finite. The new proof is short, and it is easily visualized. Received November 3, 1995; accepted in final form October 3, 1996.     

Amalgamation of Ortholattices

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.     

Orthomodular Bell-Kochen-Specker Theorem

Four elements in an orthomodular lattice of height four generate a partial Boolean subalgebra that contains a Bell-Kochen-Specker theorem. This result directly explains and generalizes the 4-dimensional Bell-Kochen-Specker theorems of various authors.