Z-mappings and a classification theorem

In this short paper, we first introduce the concept ofZ-mappings under which the image of aZ-continuous poset (respectivelyZ-algebraic poset,Z-inductive poset) is still aZ-continuous poset (respectivelyZ-algebraic poset,Z-inductive poset). We then give a classification theorem ofZ-continuous posets which generalizes an earlier work of R.-E. Hoffmann in [3]. Communicated by J. Lawson The first author thanks Professor Lawson for his help.     

Z-Join Spectra of Z-Supercompactly Generated Lattices

The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces.     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.     

Substitution and atomic extension on greedy posets

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, if P=(P ₁ , x, P ₂ )is a greedy poset, then P ₁ and P ₂ are greedy posets, the converse being false. However, for the atomic extension, P=P ₁ (x, P ₂ )is a greedy poset if and only if P ₁ and P ₂ are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containing M _n (n2), B ₃ and closed by atomic extension. The class C _n of greedy posets with jump number n is infinite. However, we show that C _n can be obtained, in a very simple way, from a subclass D _n of finite cardinal ity. We construct D _n for n=1, 2.     

How To Split Antichains In Infinite Posets*

A maximal antichain A of poset P splits if and only if there is a set B ⊂ A such that for each p ∈ P either b ≤ p for some b ∈ B or p ≤ c for some c ∈ A\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z]_P = [x,y]_P ∪ [y,z]_P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice. We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening. * This work was supported, in part, by Hungarian NSF, under contract Nos. T37846, T34702, T37758, AT 048 826, NK 62321. The second author was also supported by Bolyai Grant.     

Balance theorems for height-2 posets

We prove that every height-2 finite poset with three or more points has an incomparable pair {x, y} such that the proportion of all linear extensions of the poset in which x is less than y is between 1/3 and 2/3. A related result of Komlós says that the containment interval [1/3, 2/3] shrinks to [1/2, 1/2] in the limit as the width of height-2 posets becomes large. We conjecture that a poset denoted by V _m ⁺ maximizes the containment interval for height-2 posets of width m+1.     

Z-Semicontinuous Posets

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.     

On the automorphism conjecture for products of ordered sets

I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S ₁××S _n under the condition that max_i|S_i|=0(n/logn).     

Z-Continuous Posets and Their Topological Manifestation

A subset selection Z assigns to each partially ordered set P a certain collection Z P of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general Z-level, by replacing finite or directed sets, respectively, with arbitrary Z-sets. This leads to a theory of Z-union completeness, Z-arity, Z-soberness etc. Order-theoretical notions such as complete distributivity and continuity of lattices or posets extend to the general Z-setting as well. For example, we characterize Z-distributive posets and Z-continuous posets by certain homomorphism properties and adjunctions. It turns out that for arbitrary subset selections Z, a poset P is strongly Z-continuous iff its Z-join ideal completion Z P is Z-ary and completely distributive. Using that characterization, we show that the category of strongly Z-continuous posets (with interpolation) is concretely isomorphic to the category of Z-ary Z-complete core spaces. For suitable subset selections Y and Z, these are precisely the Y-sober core spaces.     

Verallgemeinerungen eines Satzes von S. Piccard

The following result is due to S. Piccard ([12], S.30): “If A,B ?? are Baire sets of second category and if the function f: ?×?→? is defined by f(x,y):=x?y (x,y ε ?), then the interior of f(A×B) is non void”. In this note the two main results assure, that the theorem of S. Piccard remains valid, if (1) ? is replaced by topological spaces X,Y,Z, (2) f:X×Y→Z is a function, which satisfies a certain global (respectively local) solvability condition, (3) A ?X contains a Baire set of second category and (4) B ?Y is only of second category.     

A criterion for polynomial growth of a-free two-peak posets of tame prinjective type *

It is well known from the results of L, A. Nazarova and A. G. Zavadskij [18], [19], see also [25, Chapter 15], that a poset J with one maximal element is of tame prinjective type and of polynomial growth if and only if J does not contain neither any of the Nazarova's hypercritical posets (1,1,1,1,1)^*, (1,1,1,2)^*,(2,2,3)^*, (1,3,4)^*,(W,5)^*,(1,2,6)^* nor the Nazarova-Zavadskij poset (NZ)^* (see Table 1 below). In the present paper we extend this result to a class of posets with two maximal elements. We show that Ã-free poset with two maximal elements is of tame representation type and of polynomial growth if and only if the Tits quadratic form q^s → Z (see (1.7) below) associated with J is weakly non-negative and J does not contain any of the six posets listed in Table 1 as a peak subposet.     

The Category of Directed Systems in a Category

We present two related categorical constructions. Given a category C, we construct a category C^[d], the category of directed systems in C. C embeds into C^[d], and if C has enough colimits, then C is monadic over C^[d]. Also, if E,M is a factorization structure for C, then C^[d] has a related factorization structure E_d M_d such that if E consists entirely of monic arrows, then so does E_d and the E_d-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does M_d and the M_d-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C^[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C^[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

Linear extension majority cycles in height-1 orders

For points x and y in a poset (X, >) let x> _p y mean that more linear extensions of the poset have x above y than y above x. It has been known for some time that > _p can have cycles when the height of the poset is two or more. Moreover, the smallest posets with a > _p cycle have nine points and heights of 2, 3 and 4. We show here that height-1 posets can also have > _p cycles. Our smallest example for this phenomenon has 15 points.Research supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Path-Connected Partially Ordered Sets

The graph of a partially ordered set (X, ?) has X as its set of vertices and (x,y) is an edge if and only if x covers y or y covers x. The poset is path-connected if its graph is connected. Two integer-valued metrics, distance and fence, are defined for path-connected posets. Together the values of these metrics determine a path-connected poset to within isomorphism and duality. The result holds for path-connected preordered sets where distance and fence are pseudometrics. The result fails for non-path-connected posets.     

Greedy linear extensions for minimizing bumps

Let P be a finite poset and let L={x ₁<...n} be a linear extension of P. A bump in L is an ordered pair (x _i , x _i+1) where x _ii+1 in P. The bump number of P is the least integer b(P), such that there exists a linear extension of P with b(P) bumps. We call L optimal if the number of bumps of L is b(P). We call L greedy if x _{i j} for every j>i, whenever (x _i, x _i+1) is a bump. A poset P is called greedy if every greedy linear extension of P is optimal. Our main result is that in a greedy poset every optimal linear extension is greedy. As a consequence, we prove that every greedy poset of bump number k is the linear sum of k+1 greedy posets, each of bump number zero.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

On Edge Decompositions of Posets

An edge decomposition of a poset P is a collection of chains such that every pair of elements of which one covers the other belongs to exactly one chain. We consider this and the related notion of the line poset L(P) which consists of pairs of adjacent elements of P so that (xy)<_L(P) (x'y') iff y _P x'. We present some min-max type results on path-cycle partitions of digraphs which are applicable to poset decompositions. Providing an explicit construction we show that the lattice of the subsets of an n-set admits an edge decomposition into symmetric chains. We demonstrate a few applications of this decomposition. Also, a characterisation of line posets is given.     

Greedy posets for the bump-minimizing problem

A bump (x _i,x _i+1) occurs in a linear extension L={x ₁<...n} of a poset P, if x _ii+1 in P. L. is greedy if x _ij for every j>i, whenever (x _i x _i+1) in a bump in L. The purpose of this paper is to give a characterization of all greedy posets. These are the posets for which every greedy linear extension has a minimum number of bumps.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

3-Interval irreducible partially ordered sets

In this paper we discuss the characterization problem for posets of interval dimension at most 2. We compile the minimal list of forbidden posets for interval dimension 2. Members of this list are called 3-interval irreducible posets. The problem is related to a series of characterization problems which have been solved earlier. These are: The characterization of planar lattices, due to Kelly and Rival [5], the characterization of posets of dimension at most 2 (3-irreducible posets) which has been obtained independently by Trotter and Moore [8] and by Kelly [4] and the characterization of bipartite 3-interval irreducible posets due to Trotter [9].We show that every 3-interval irreducible poset is a reduced partial stack of some bipartite 3-interval irreducible poset. Moreover, we succeed in classifying the 3-interval irreducible partial stacks of most of the bipartite 3-interval irreducible posets. Our arguments depend on a transformationP B(P), such that IdimP=dimB(P). This transformation has been introduced in [2].Supported by the DFG under grant FE 340/2–1.     

广义度量空间和偏序集

广义度量空间和偏序集都具有函数空间．而函数空间的存在为数学构造和计算提供了很大方便．本文还讨论了广义度量空间和偏序集之间的相互转化问题．