Order

We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic to direct products of basic BCK-chains, generalizing a result of Cignoli for MV-algebras.

Consider two partially ordered setsP, Q and a number of edges connecting some of the points ofP with some of the points ofQ. This yields a bipartite graph. Some pairs of the edges may cross each other because their endpoints atP andQ are oppositely ordered. A natural decrossing operation is to exchange the endpoints of these edges incident atQ, say. This is called a switch. A left lift of an edge means to replace its starting point atP by a larger starting point. A right lift is defined symmetrically for the endpoints atQ. The operation of adding an edge cannot, informally, be explained better. Assume we are given two bipartite graphs π, σ on the node setPσQ. We show that for certain pairs (P, Q) of finite posets, a neat necessary and sufficient criterion can be given in order that σ is obtainable from π by the sequence of elementary operations just defined. A recent characterization of the Bruhat order of the symmetric group follows as a special case.

Throughout much of the literature in economics and political science, the notion of separability provides a mechanism for characterizing interdependence within individual preferences over multiple dimensions. In this paper, we show how preseparable extensions can be used to construct certain classes of separable and non-separable preferences. We prove several associated combinatorial results, and we note a correspondence between separable preference orders, Boolean term orders, and comparative probability relations. We also mention several open questions pertaining to preseparable extensions and separable preferences.

The symmetric group $\mathfrak {S}_{n}$ acts naturally on the poset of injective words over the alphabet {1, 2,…,n}. The induced representation on the homology of this poset has been computed by Reiner and Webb. We generalize their result by computing the representation of $\mathfrak {S}_{n}$ on the homology of all rank-selected subposets, in the sense of Stanley. A further generalization to the poset of r-colored injective words is given.

The following result is proved in this note: For any positive integers w and t, if an ordered set P has jump number at least (t+1) w−1, then either the width of P is more than w, or P has a tower, i.e., a linear sum of pairs of noncomparable elements, of height more than t.

Given a set of tasks with precedence constraints, each of them requiring one unit of time to be completed by a processor, what are the hidden structures which prescribe the minimal completion time? To deal with this question, we introduce a notion of minimality for partial orders on a set of tasks. We characterize these minimal orders in the case of two processors. Another notion of maximality leads to a new proof (communicated by M. Pouzet) of a classic result in two-processor scheduling.

In this paper, we introduce a binary relation on the vertex set of a k-tournament, and using this relation show that every finite poset with cardinality n≥4 can be represented by a k-tournament for every k satisfying 3≤k≤n−1.

Z-semicontinuous posets include semicontinuous lattices and Z-continuous posets as special cases. We characterized when the associated Z-waybelow relation is multiplicative and also make a topological connection.