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We prove that the Katětov order on Borel ideals (1) contains a copy of 𝓟 ( ω ) / Fin $\mathcal {P}(\omega )/\mathbf {Fin}$ , consequently it has increasing and decreasing chains of lenght 𝔟; (2) the sequence Finα (α < ω1) is a strictly increasing chain; and (3) in the Cohen model, Katětov order does not contain any increasing...
We prove a generalization of Debreu’s Open Gap Lemma. Given any subset of the real line, this lemma guarantees the existence of a strictly increasing real function such that all the gaps of the image of the subset are open. Now we extend it to the case of n subsets of the real line and study the existence of a strictly increasing real function such that all the gaps of the image of each set are open...
A generalized pseudo effect algebra (GPEA) is a partially ordered partial algebraic structure with a smallest element 0, but not necessarily with a unit (i.e, a largest element). If a GPEA admits a so-called unitizing automorphism, then it can be embedded as an order ideal in its so-called unitization, which does have a unit. We study unitizations of GPEAs with respect to a unitizing automorphism,...
A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of slim semimodular lattices of size n is asymptotically C · 2n.
We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern p is denoted AvP(p). We extend a proof of Simion and Schmidt to show that AvP(132)=Av ...
For elements x and y in the (Hasse) diagram D of a finite bounded poset P, x is on the left of y, written as xλy, if x and y are incomparable and x is on the left of all maximal chains through y. Being on the right, written as xϱy, is defined analogously. The diagram D is quasiplanar if λ and ϱ are transitive and for any pair (x,y) of incomparable elements,...
Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S5 as a subposet, then the cover graph of P has treewidth at least...
We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every ℝ n -valued σ-additive modular measure on a σ-complete pseudo-D-lattice.
We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <1,...,<r be partial orders on P. Let G(P, <1,...,<r) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x<iy or y<ix holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <1, ... , <r) is strictly larger than...
In this paper, we weaken the conditions for the existence of adjoint closure operators, going beyond the standard requirement of additivity/co-additivity. We consider the notion of join-uniform (lower) closure operators, introduced in computer science, in order to model perfect lossless compression in transformations acting on complete lattices. Starting from Janowitz’s characterization of residuated...
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes. We then exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, $\mathcal {D}(G;\mathcal {F})$ whose faces are vertex subsets of G that induce ...
Generalized Esakia spaces are the topological duals of bounded implicative semilattices in the duality studied by G. Bezhanishvili and R. Jansana. We study the relation between a Hilbert algebra and the generalized Esakia space dual to its free implicative semilattice extension. To establish the relation we introduce a category whose objects are a generalized Esakia space together with a family of...
Let κ = 2ω, and assume f : ℝ → P ( ℝ ) $f:\mathbb {R}\rightarrow \mathcal {P}(\mathbb {R})$ satisfies the intersection properties C(ω,κ) and C(κ,ω). We prove that if 𝔯 < cf ( κ ) $\mathfrak {r}<\text {cf}(\kappa )$ then there exists a dense free set for f.
In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congruence-simple atomistic lattices is generalized. We prove that the result remains valid for any finite tolerance-simple lattice; moreover, we extend it to a type of subdirect product of such lattices, introducing the notion of a generalized oligarchy.
This paper clarifies the connection between multiple criteria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical formulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent...
We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors N intervals of the real line using Θ ( log N / k ) ${\Theta }(\log N/k)$ colors such that for every point p, contained in at least k intervals, not all the intervals containing p have the same color. We also prove the corresponding result about online...
This paper investigates the class of ordered sets that are embeddable into a distributive lattice in such a way that all existing finite meets and joins are preserved. The main result is that the following decision problem is NP-complete: Given a finite ordered set, is it embeddable into a distributive lattice with preservation of existing meets and joins? The NP-hardness of the problem is proved...
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