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In this note we want to point out the usefulness of antitone operations in the study of games on graphs. We demonstrate this by proving, among others, the generalizations from games to antitone operations of theorems stated by Fraenkel. This perspective makes also clear the general nature of some problems by separating what is general and what is particular.
Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2X generated by F, let G be a Hausdorff commutative topological lattice group and let rbaF(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rbaF(A(F), G), we extend to this setting the first decomposition theorem...
Congruences and ideals in partial Abelian monoids (PAM) are studied. It is shown that the so-called R1-ideals in cancellative PAMs (CPAM) form a complete Brouwerian sublattice of the lattice of all ideals, and they are standard elements of it. In a special class of CPAMs, effect algebras, properties of ideals and congruences are studied in relation to the generalized Sasaki projections and dimensional...
We investigate the structure of m-jump-critical posets P with w(P) = m. We prove that the size of such posets satisfies |P| ≤ 3m2. For the special when the maximum antichain occurs as the maximal (or minimal) elements, we have the sharp upper bound |P| ≤ 3m - k; where k = min {|{Max}(P)|, |{Min}(P)|}. We give examples of posets which illustrate the explored structure of these m-jump-critical posets...
Initial chain algebras on pseudotrees generalize the notion of an interval algebra on a linear order. Many relationships which hold between the various cardinal functions on interval algebras also hold for initial chain algebras. In particular, for initial chain algebras on pseudotrees, depth equals tightness, spread equals hereditary Lindelöf degree, irredundance equals the cardinality of the algebra,...
We study Dedekind complete commutative BCK-algebras with the relative cancellation property and their connection with corresponding universal groups. We shall characterize Dedekind orthogonally complete atomic and Archimedean BCK-algebras, generalizing results of Jakubík known for MV-algebras. Finally, we characterize those Dedekind complete and atomic commutative BCK-algebras that are isomorphic...
Generalizing the proof of the theorem describing the closed cone of flag f-vectors of arbitrary graded posets, we give a description of the cone of flag f-vectors of planar graded posets. The labeling used is a special case of a “chain-edge labeling with the first atom property”, or FA-labeling, which also generalizes the notion of lexicographic shelling, or CL-labeling. The resulting analogy suggests...
There is a natural way to associate with a poset P a hypergraph HP, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP. The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially less than HP. We give a new proof of the fact that the dimension of P is 2 if and only if GP is bipartite...
The recognition complexity of ordered set properties is considered in terms of how many questions must be put to an adversary to decide if an unknown partial order has the prescribed property. We prove a lower bound of order n2 for properties that are characterized by forbidden substructures of fixed size. For the properties being connected, and having exactly k comparable pairs, k ≤ n2 / 4 we show...
We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum...
We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.
O*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F*-rings. We show that the two classes of rings coincide.
It is proved that the collection of all finite lattices with the same partially ordered set of meet-irreducible elements can be ordered in a natural way so that the obtained poset is a lattice. Necessary and sufficient conditions under which this lattice is Boolean, distributive and modular are given.
Given a k-tuple P=(x1,x2,...,xk) in a finite lattice X endowed with the lattice metric d, a median of P is an element m of X minimizing the sum ∑id(m,xi). If X is an upper semimodular lattice, Leclerc proved that a lower bound of the medians is c(P), the majority rule and he pointed out an open problem: “Is c1(P)=∨ixi, the upper bound of the medians?” This paper shows that the upper...
We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.
We show that the neighborhood deck of an ordered set can be reconstructed from the deck of one point deleted subsets. As a consequence of the above results we reconstruct some maximal cards and present short new proofs of the reconstructibility of ordered sets of width 2 and of the recognizability of N-free ordered sets. We also reconstruct the maximal cards of N-free ordered sets.
We compute the subdirectly irreducible factors of the free Hermitean (semi)lattice generated by a single element a subject to the relation a≤a⊥; we also compute the factors of its induced distributive presentation.
Erhlich introduced the concept of generating combinatorial structures in constant time per generated item. Such algorithms are called “loopless” and have been described for many objects. Myers introduced the idea of a basic minimal interval order. This paper presents a loopless algorithm for generating basic minimal interval orders.
Every model of ZFC contains a product of two linear orders, each of size ℵ1, with the property that every subset or complement thereof contains a maximal chain.
It is proved that the C-core of a chain-complete ordered set is unique up to isomorphism if it exists. We also give an example that shows that the (UC∪LC)-core of an ordered set need not be unique. This is related to a question, which asks if the P-core is unique if it exists.
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