Abstract: We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple conditions defining orthodomains and the additional requirement of having enough directions. Excepting pathologies involving maximal Boolean subalgebras of four elements, it is shown that there is an equivalence between the category of orthoalgebras and the category of orthodomains with enough directions with morphisms suitably defined. Furthermore, we develop a representation of orthodomains with enough directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph approach extends the technique of Greechie diagrams and resembles projective geometry. Using such hypergraphs, every orthomodular poset can be represented by a set of points and lines where each line contains exactly three points. PubDate: 2019-03-09

Abstract: We define and study an alternative partial order, called the spectral order, on a synaptic algebra—a generalization of the self-adjoint part of a von Neumann algebra. We prove that if the synaptic algebra A is norm complete (a Banach synaptic algebra), then under the spectral order, A is Dedekind σ-complete lattice, and the corresponding effect algebra E is a σ-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh algebra in both the usual (synaptic) and spectral ordering; and if A is Banach, then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite type, then De Morgan laws hold on E in both the synaptic and spectral ordering. PubDate: 2019-03-01

Abstract: Schimmerling asked whether \(\square ^{\ast }_{\lambda }+\mathsf {GCH}\) entails the existence of a λ+-Souslin tree, for a singular cardinal λ. We provide an affirmative answer under the additional assumption that there exists a non-reflecting stationary subset of \({E}^{{\lambda }^{+}}_{{\neq } \text {cf}(\lambda )}\) . As a bonus, the outcome λ+-Souslin tree is moreover free. PubDate: 2019-02-21

Abstract: In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such orders and the infinite reduced words. We show that for a 2 closure biclosed set B in Φ+, the B-twisted weak order is a non-complete meet semilattice if B is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups. PubDate: 2019-01-03

Abstract: In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset has a nested chain decomposition. This conjecture is proved to be true only for some posets of small ranks (Wang Discrete Math. 145(3), 493–497, 2005; Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011). In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfy the Griggs’s conjecture by refining the proofs in the two papers (Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011). PubDate: 2018-12-04

Abstract: In this paper for t > 2 and n > 2, we give a simple upper bound on a ([t]n), the number of antichains in chain product poset [t]n. When t = 2, the problem reduces to classical Dedekind’s problem posed in 1897 and studied extensively afterwards. However few upper bounds have been proposed for t > 2 and n > 2. The new bound is derived with straightforward extension of bracketing decomposition used by Hansel for bound \(3^{n\choose \lfloor n/2\rfloor }\) for classical Dedekind’s problem. To our best knowledge, our new bound is the best when \({\Theta }\left (\left (\log _{2}t\right )^{2}\right )=\frac {6t^{4}\left (\log _{2}\left (t + 1\right )\right )^{2}}{\pi \left (t^{2}-1\right )\left (2t-\frac {1}{2}\log _{2}\left (\pi t\right )\right )^{2}}<n\) and \(t=\omega \left (\frac {n^{1/8}}{\left (\log _{2}n\right )^{3/4}}\right )\) . PubDate: 2018-12-04

Abstract: Let F be a non-archimedean o-field and C be the field of generalized complex numbers over F. In this paper, we describe all the directed partial orders on C with 1 > 0 by using admissible semigroups of F +. PubDate: 2018-11-01

Abstract: For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras. PubDate: 2018-11-01

Abstract: We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient. PubDate: 2018-11-01

Abstract: We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product. PubDate: 2018-11-01

Abstract: Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x1x2⋯xn in one-line notation. For distinct elements x, y ∈ P, we define ℙ(x ≺ y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\) , we say (x, y) is an α-balanced pair if α ≤ ℙ(x ≺ y) ≤ 1 − α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed. PubDate: 2018-11-01

Abstract: The higher Bruhat order is a poset generalizing the weak order on permutations. Another special case of this poset is an ordering on simple wiring diagrams. For this case, we prove that every interval is either contractible or homotopy equivalent to a sphere. This partially proves a conjecture due to Reiner. Our proof uses some tools developed by Felsner and Weil to study wiring diagrams. PubDate: 2018-11-01

Abstract: Let f(n) be the largest integer such that every poset on n elements has a 2-dimensional subposet on f(n) elements. What is the asymptotics of f(n)' It is easy to see that f(n) = n 1/2. We improve the best known upper bound and show f(n) = O (n 2/3). For higher dimensions, we show \(f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )\) , where f d (n) is the largest integer such that every poset on n elements has a d-dimensional subposet on f d (n) elements. PubDate: 2018-11-01

Authors:F. Mwesigye; J. K. Truss Abstract: Two structures A and B are n-equivalent if player II has a winning strategy in the n-move Ehrenfeucht-Fraïssé game on A and B. We extend earlier results about n-equivalence classes for finite coloured linear orders, describing an algorithm for reducing to canonical form under 2-equivalence, and concentrating on the cases of 2 and 3 moves. PubDate: 2018-05-02 DOI: 10.1007/s11083-018-9458-3

Authors:Shimon Garti Abstract: Let κ be a successor cardinal. We prove that consistently every bipartite graph of size κ+ × κ+ contains either an independent set or a clique of size τ × τ for every ordinal τ < κ+. We prove a similar theorem for ℓ-partite graphs. PubDate: 2018-03-26 DOI: 10.1007/s11083-018-9457-4

Authors:Antonio Di Nola; Giacomo Lenzi; Anna Carla Russo Abstract: In this paper we study some invariants for MV-algebras and thanks to Mundici’s equivalence we transfer these invariants to ℓ-groups with strong unit. In particular, we prove that, as it happens to MV-algebras, every ℓ-u group has two families of skeletons, which we call the n-skeletons and the \({}_{n}^{\omega }\) -skeletons. Then we study the classes of ℓ-u groups (and of MV-algebras) which coincide with the union of such skeletons, called here ω-skeletal and \({}_{\omega }^{\omega }\) -skeletal ℓ-u groups (resp. MV-algebras). We also analyze the problem of axiomatizing in terms of geometric theories or theories of presheaf type these classes of ℓ-u groups (and of MV-algebras). PubDate: 2018-03-23 DOI: 10.1007/s11083-018-9456-5

Authors:Ralph Freese; J. B. Nation; Matt Valeriote Abstract: This paper investigates the computational complexity of deciding if a given finite algebra is an expansion of a semilattice. In general this problem is known to be EXP-TIME complete, and we show that even for idempotent algebras, this problem remains hard. This result is in contrast to a series of results that show that similar decision problems turn out to be tractable. PubDate: 2018-03-14 DOI: 10.1007/s11083-018-9455-6

Authors:Erkko Lehtonen; Tamás Waldhauser Abstract: We study the structure of the partially ordered set of minors of an arbitrary function of several variables. We give an abstract characterization of such “minor posets” in terms of colorings of partition lattices, and we also present infinite families of examples as well as some constructions that can be used to build new minor posets. PubDate: 2018-03-10 DOI: 10.1007/s11083-018-9453-8

Authors:Dániel T. Soukup Abstract: We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f(x) ≤ f(y). We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of R. Camerlo, R. Carroy and A. Marcone. In particular, ♢+ implies the existence of a lexicographically ordered Suslin tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under \(2^{\aleph _{0}}<2^{\aleph _{1}}\) or in the Cohen and other canonical models (where \(2^{\aleph _{0}}= 2^{\aleph _{1}}\) ); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. PubDate: 2018-02-21 DOI: 10.1007/s11083-018-9454-7

Authors:Lorenzo Carlucci Abstract: Recent results of Hindman, Leader and Strauss and of Fernández-Bretón and Rinot showed that natural versions of Hindman’s Theorem fail for all uncontable cardinals. On the other hand, Komjáth proved a result in the positive direction, showing that there are arbitrarily large abelian groups satisfying some Hindman-type property. In this note we show how a family of natural Hindman-type theorems for uncountable cardinals can be obtained by adapting some recent results of the author from their original countable setting. PubDate: 2018-01-24 DOI: 10.1007/s11083-018-9452-9