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A nearlattice A is a meet semilattice in which every initial segment is a lattice; A is said to be Boolean if all of these lattices are Boolean. Overriding on A is a binary operation defined by $ a \triangleleft {b} := \sup\{x\colon (x \le a \mbox{ and } x {\mathrel{\mbox{\,\raisebox{.7ex}{$\scriptstyle|$}\hspace{-.75ex}\raisebox{-.35ex}{$\circ$}}}}b) \mbox{ or } x \le b\} $...
Distributive lattices are well known to be precisely those lattices that possess cancellation: and imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M3 or N5 as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the...
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger...
We consider tournaments defined by several linear orderings of the vertex set according to some rule specifying the direction of each arc depending only on the order of the end vertices in each of the orderings. In the finite case, it was proved in Alon et al. (J Comb Theory Ser B 96:374–387, 2006) that the domination number of such tournaments is bounded in terms of the rule only. We show that for...
We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the...
We completely determine all distributive, codistributive, standard, costandard, and neutral elements in the lattice of overcommutative semigroup varieties, thus correcting a gap in a previous paper.
Given a tournament T = (X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper...
The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n ⩾ 1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this...
This paper studies an extension of biorders that has a “frontier” between the relation and the absence of relation. This extension is motivated by a conjoint measurement problem consisting in the additive representation of ordered coverings defined on product sets of two components. We also investigate interval orders and semiorders with frontier.
For a finite poset P = (V, ≤ ), let ${\cal B}_s(P)$ consist of all triples (x,y,z) ∈ V3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let ${\cal B}_s(G)$ consist of all triples (x,y,z) ∈ V3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations ${\cal B}_s(P)$ and...
We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991;...
Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting...
The d-dimensional random partial order is the intersection of d independently and uniformly chosen (with replacement) linear orders on the set [n] = {1, 2, . . . , n}. This is equivalent to picking n points uniformly at random in the d-dimensional unit cube with the coordinate-wise ordering. If d = 2, then this can be rephrased by declaring that for any pair P1, P2 ∈ Q2 we have...
A class of posets, called thin posets, is introduced, and it is shown that every thin poset can be covered by a finite family of trees. This fact is used to show that (within ZFC) every separable monotonically Menger space is first countable. This contrasts with the previously known fact that under CH there are countable monotonically Lindelöf spaces which are not first countable.
We generalize the notion of saturated orders to infinite partial orders and give both a set-theoretic and an algebraic characterization of such orders. We then study the proof theoretic strength of the equivalence of these characterizations in the context of reverse mathematics, showing that depending on one’s choice of definitions, this equivalence is either provable in RCA 0 or equivalent to ACA...
In 1975, J. Griggs conjectured that a normalized matching rank-unimodal poset possesses a nested chain decomposition. This elegant conjecture remains open even for posets of rank 3. Recently, Hsu, Logan, and Shahriari have made progress by developing techniques that produce nested chain decompositions for posets with certain rank numbers. As a demonstration of their methods, they prove that the conjecture...
In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication...
The category of finite directed graphs is Cartesian closed, hence it has a product and exponential objects. For a fixed K, let be the class of all directed graphs of the form KG, preordered by the existence of homomorphisms, and factored by homomorphic equivalence. It has long been known that is always...
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