Showing papers in "Order in 1985"
TL;DR: In this article, the authors define geometric semilattices, a generalization of geometric lattices, and prove several axiomatic and constructive characterizations, for example, the poset of independent sets of a matroid.
Abstract: We define geometric semilattices, a generalization of geometric lattices. The poset of independent sets of a matroid is another example. We prove several axiomatic and constructive characterizations, for example: geometric semilattices are those semilattices obtained by removing a principal filter from a geometric lattice. We also show that all geometric semilattices are shellable, unifying and extending several previous results.
75 citations
66 citations
TL;DR: In this paper, the authors investigated the lattice Co(P) of convex subsets of a general partially ordered set P and gave necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(p) for some P.
Abstract: The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.
63 citations
TL;DR: In this paper, a tensor product for complete lattices via concept lattices is studied and a characterization as a universal solution and an ideal representation of the tensor products are given.
Abstract: A tensor product for complete lattices is studied via concept lattices. A characterization as a universal solution and an ideal representation of the tensor products are given. In a large class of concept lattices which contains all finite ones, the subdirect decompositions of a tensor product can be determined by the subdirect decompositions of its factors. As a consequence, one obtains that the tensor product of completely subdirectly irreducible concept lattices of this class is again completely subdirectly irreducible. Finally, applications to conceptual measurement are discussed.
52 citations
TL;DR: A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width for solving a more general weighted problem in precedence constrained scheduling.
Abstract: A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.
49 citations
TL;DR: In this paper, the authors studied some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set and showed that x2 is not bounded by a function of the girth of the graph.
Abstract: We study some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set. For each κ≥1, M. Aigner and G. Prins have introduced a notion of a vertex colouring, here called κ-good colouring, such that a 1-good colouring is the usual concept and graphs that have a 2-good colouring are precisely covering graphs. We present some inequalities for the corresponding chromatic numbers χκ, especially for x2. There exist graphs that satisfy these inequalities for κ=2 but are not covering graphs. We show also that x2 cannot be bounded by a function of x=x1. A construction of Nesetřil and Rodl is used to show that x2 is not bounded by a function of the girth.
35 citations
TL;DR: The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive Lattice with a compact unit has a prime element as mentioned in this paper, which is then used to deduce from PIT that every nontrivial ring with unit and every Wallman locale is spatial.
Abstract: The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive lattice with a compact unit has a prime element, which is then used to deduce from PIT that (1) every nontrivial ring with unit has a prime ideal, and (2) every Wallman locale is spatial.
23 citations
TL;DR: The class of orthomodular lattices which have only finitely many commutators was investigated in this article, and the following theorems were proved: every irreducible element of the class is a direct product of a Boolean algebra.
Abstract: The class \(\mathfrak{C}\) of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: \(\mathfrak{C}\) contains the block-finite orthomodular lattices. Every irreducible element of \(\mathfrak{C}\) is simple. Every element of \(\mathfrak{C}\) is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of \(\mathfrak{C}\) which are modular, or are M-symmetric with at least one atom, have height two or less.
22 citations
TL;DR: In this paper, it was shown that in a finite consistent lattice, the incidence matrix of meet-irreducible versus joinirreduceible elements has rank the number of join-irereducibles.
Abstract: An element in a lattice is join-irreducible if x=a∨b implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element x∨j is a join-irreducible in the upper interval [x, i]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.
19 citations
TL;DR: In this paper, it was shown that the fixed point property is comparability invariant for finite ordered sets with isomorphic comparability graphs, i.e., if P and Q are two sets and Q does not have a comparability graph, then P has the fixed-point property if and only if Q does.
Abstract: We show that the fixed point property is comparability invariant for finite ordered sets; that is, if P and Q are finite ordered sets with isomorphic comparability graphs, then P has the fixed point property if and only if Q does. In the process we give a characterization of comparability invariants which can also be used to give shorter proofs of some known results.
19 citations
TL;DR: In this article, it was shown that if ℒ is a finite lattice, and r is an integral valued function on the lattice satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice I containing ∆ as a sublattice such that r is the height function of ∆ restricted to ∆.
Abstract: In this paper we prove that if ℒ is a finite lattice, and r is an integral valued function on ℒ satisfying some very natural conditions, then there exists a finite geometric (that is, semimodular and atomistic) lattice I containing ℒ as a sublattice such that r is the height function of ℒ restricted to ℒ. Moreover, we show that if, for all intervals [e, f] of ℒ, semimodular lattices I, of length at most r(f)-r(e) are given, then I can be chosen to contain I in its interval [e, f] as a cover preserving {0}-sublattice. As applications, we obtain results of R. P. Dilworth and D. T. Finkbeiner.
TL;DR: In this paper, a minimum cardinality family of n-uniform, k-edge hypergraphs satisfying the divisibility condition has been determined, except for finitely many k-decomposition families.
Abstract: We determine a minimum cardinality family ℱn, k (resp. ℋn, k) ofn-uniform,k-edge hypergraphs satisfying the following property: all, except for finitely many,n-uniform hypergraphs satisfying the divisibility condition have an ℱn, k-decomposition (resp. vertex ℋn, k-decomposition).
TL;DR: In this paper, the authors proved that the poset scheduling problem is NP-hard and proved that it is transformable to the closed set problem or the minimum cut problem in a network.
Abstract: Let P and Q be two finite posets and for each p∈P and q∈Q let c(p, q) be a specified (real-valued) cost. The poset scheduling problem is to find a function s: P→Q such that Σ
p∈P
c(p,s(p)) is minimized, subject to the constraints that p
TL;DR: In this article, the authors introduced the concept of greedy dimension for partially ordered sets and proved that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.
Abstract: This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.
TL;DR: In this article, it was shown that for each fixed k>1, with probability approaching 1 as n→∞, the comparability graph of a set of n points chosen randomly and independently from the unit cube in ℝ�ℝÃÂÃÂÃÂÃÂÃÂÃÂÃÂÃÂk€€€ Á€ À€ Þ with the induced product order is connected and has diameter 3.
Abstract: Let P
k
(n) be the (partial) order determined by intersecting k random linear orderings of a set of size n; equivalently, let P
k
(n) consist of n points chosen randomly and independently from the unit cube in ℝ
k
, with the induced product order We show for each fixed k>1, that with probability approaching 1 as n→∞, the comparability graph of P
k
(n) is connected and has diameter 3
TL;DR: In this article, sufficient conditions for the fixed point property of products of two partially ordered sets are proved in terms of multifunctions (functions with non-empty sets as values).
Abstract: Sufficient conditions for the fixed point property for products of two partially ordered sets are proved. These conditions are formulated in terms of multifunctions (functions with non-empty sets as values).
TL;DR: The first of a planned series of papers on the structure of non-Arguesian modular lattices has been published in this paper, where it was shown that there exist points and lines in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a "classical" non-arguesian configuration.
Abstract: This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by ‘badly’ gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a ‘classical’ non-Arguesian configuration.
TL;DR: In this article, a decomposition approach for the jump number problem was proposed from an algorithmic point of view, based on which some new classes of partial orders are identified, for which the problem is polynomially solvable.
Abstract: Consider the linear extensions of a partial order. A jump occurs in a linear extension if two consecutive elements are unrelated in the partial order. The jump number problem is to find a linear extension of the ordered set which contains the smallest possible number of jumps. We discuss a decomposition approach for this problem from an algorithmic point of view. Based on this some new classes of partial orders are identified, for which the problem is polynomially solvable.
TL;DR: The partially ordered set P is an (α, β, γ) ordered set if the width of P, the length of any chain of P and the cut-set number ≤γ as discussed by the authors.
Abstract: The partially ordered set P is an (α, β, γ) ordered set if the width of P≥α, the length of any chain of P≤β and the cut-set number ≤γ. We will prove that if P is an (α, β, γ) ordered set then P contains a ‘simple’ (α, β, γ) ordered set and use this result to prove that if P has the 3 cutset property, then width of P ≤ length of P+3.
[...]
TL;DR: For a finite poset P and x, N Linial as mentioned in this paper showed that the bound 1/3 cannot be further increased to 2/3 for a poset of width 2.
Abstract: For a finite poset P and x, yeP let pr(x>y) be the fraction of linear extensions which put x above y N Linial has shown that for posets of width 2 there is always a pair x, y with 1/3 ⩽ pr(x>y)⩽2/3 The disjoint union C
1∪C
2 of a 1-element chain with a 2-element chain shows that the bound 1/3 cannot be further increased In this paper the extreme case is characterized: If P is a poset of width 2 then the bound 1/3 is exact iff P is an ordinal sum of C
1∪C
2's and C
1's
TL;DR: In this paper, the authors consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semi-attices, and they show that a continuous semilax with an infinite anticheain converging to a larger element contains a semilectice copy of Δ.
Abstract: In this paper we consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semilattices. Conditions are first considered that lead to the antichain generating a copy of a countable product of the two-element semilattice. Then a special semilattice, called Δ, is defined, its basic properties developed, and it is shown in our main result that a continuous semilattice with an infinite antichain converging to a larger element contains a semilattice copy of Δ. The paper closes with a consideration of countable antichains that converge to a lower element or a parallel element and the kinds of semilattices generated in this context.
TL;DR: In this paper, the structure of normal subgroup lattices of 2-transitive automorphism groups of infinite linearly ordered sets (Ω, ≤) was examined using combinatorial and model-theoretic means.
Abstract: Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A(Ω)) of 2-transitive automorphism groups A(Ω) of infinite linearly ordered sets (Ω, ≤). Certain natural sublattices of N(A(Ω)) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion \((\bar \Omega , \leqslant )\) of (Ω, ≤). As a consequence, A(Ω) has either precisely 5 or at least 22ℵ1 (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.
TL;DR: In this paper, it was shown that for a lattice L with a glued tolerance relation, the number of elements of L with exactly k lower (upper) covers is constant.
Abstract: A tolerance relation θ of a lattice L, i.e., a reflexive and symmetric relation of L which is compatible with join and meet, is called glued if covering blocks of θ have nonempty intersection. For a lattice L with a glued tolerance relation we prove a formula counting the number of elements of L with exactly k lower (upper) covers. Moreover, we prove similar formulas for incidence structures and graphs and we give a new proof of Dilworth's covering theorem.
TL;DR: In this article, it was shown that if P is a poset containing noN, then every minimal cutset in P is an antichain, and that the converse also holds when P is finite.
Abstract: It is shown that ifP is a poset containing noN, then every minimal cutset inP is an antichain, that the converse also holds whenP is finite, and that this converse fails in general.
TL;DR: In this paper, it was shown that all C4-free graphs and all covering graphs of finite lattices are strongly sensitive to lattice isomorphisms of a graph G onto a given graph G.
Abstract: Let ℒ(G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism Φ: ℒV(G)≃ℒ(G′) induces a bijection ϕ: V(G)→V(G′) such that for each x in V(G), ϕ(x)=x' in V(G') iff Φ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism Φ: ℒ(G)≃ℒ(G′), the bijection ϕ induced by Φ is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C4-free graphs and all covering graphs of finite lattices are strongly sensitive.
TL;DR: In this article, it was shown that there is a matroid corresponding to any N-free poset and apply the Rado-Edmonds Theorem to obtain another proof of this result.
Abstract: Let L=x1x2...xn be a linear extension of a poset P. Each pair (xi, xi+1) such that xi≮ xi+1in P is called a jump of L. It is well known that for N-free posets a natural ‘greedy’ procedure constructing linear extensions yields a linear extension with a minimum number of jumps. We show that there is a matroid corresponding to any N-free poset and apply the Rado-Edmonds Theorem to obtain another proof of this result.
TL;DR: In this article, a subset of P containing no l+1 elements which are identical in M−1 components and linearly ordered in the Mth one is considered. But the result is not applicable to all P. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.
Abstract: Let P=P1×P2×...×PM be the direct product of symmetric chain orders P1, P2, ..., PM. Let F be a subset of P containing no l+1 elements which are identical in M−1 components and linearly ordered in the Mth one. Then max |F|≤c•M1/2•l•W(P), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and l. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.
TL;DR: In this paper, it was shown that a poset cannot be isomorphic to the system of all lower ends (i.e., directed lower ends) only if every ideal is principal.
Abstract: A standard extension for a poset P is a system Q of lower ends (‘descending subsets’) of P containing all principal ideals of P. An isomorphism ϕ between P and Q is called recycling if ∪ϕ[Y]∈Q for all Y∈Q. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.
TL;DR: In this paper, the greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. The greedy dimension does not exceed the width of P and |P−A|≥2.
Abstract: Every linear extension L: [x1