Telgársky’s conjecture states that for each k ∈ ℕ, there is a topological space Xk such in the Banach-Mazur game on Xk, player nonempty has winning (k + 1)-tactic but no k-tactic. We prove this statement consistently false. More specifically, we prove, assuming GCH+□, if strategy T3 X, then she 2-tactic. The proof uses coding argument due to Galvin, whereby X π-base with certain nice properties...

A subset S of a poset (partially ordered set) is convex if and only if S contains every poset element which is between any two elements in S. Poset convex subsets arise in applications that involve precedence constraints, such as in project scheduling, production planning, and assembly line balancing. We give a strongly polynomial time algorithm which, given a poset and element weights (of arbi...

Given a relation R ⊆ O × A on a set O of objects and a set A of attributes, the Galois sub-hierarchy (also called AOC-poset) is the partial order on the introducers of objects and attributes in the corresponding concept lattice. We present a new efficient algorithm for building a Galois sub-hierarchy which runs in O(min{nm, nα}), where n is the number of objects or attributes, m is the size of ...

in this paper we study the notions of cogenerator and subdirectlyirreducible in the category of s-poset. first we give somenecessary and sufficient conditions for a cogenerator $s$-posets.then we see that under some conditions, regular injectivityimplies generator and cogenerator. recalling birkhoff'srepresentation theorem for algebra, we study subdirectlyirreducible s-posets and give this theo...

In this paper we study the notions of cogenerator and subdirectlyirreducible in the category of S-poset. First we give somenecessary and sufficient conditions for a cogenerator $S$-posets.Then we see that under some conditions, regular injectivityimplies generator and cogenerator. Recalling Birkhoff'sRepresentation Theorem for algebra, we study subdirectlyirreducible S-posets and give this theo...

We study flag enumeration in intervals in the uncrossing partial order on matchings. We produce a recursion for the cd-indices of intervals in the uncrossing poset Pn. We explicitly describe the matchings by constructing an order-reversing bijection. We obtain a recursion for the ab-indices of intervals in the poset P̂n, the poset Pn with a unique minimum 0̂ adjoined.

(1) Let R be a division ring. Prove that every module over R is free. You will need to use Zorn’s lemma: Recall that a partially order set (=poset) S is a set with a relation x ≤ y defined between some pairs of elements x, y ∈ S, such that: (i) x ≤ x ; (ii) x ≤ y and y ≤ x implies x = y ; (iii) x ≤ y , y ≤ z ⇒ x ≤ z . A chain in S is a subset T ⊂ S such that for all t, t ′ in T , either t ≤ t ′...

For a poset P and a distributive 〈∨, 0〉-semilattice S, a S-valued poset measure on P is a map μ : P×P → S such that μ(x, z) ≤ μ(x, y)∨μ(y, z), and x ≤ y implies that μ(x, y) = 0, for all x, y, z ∈ P . In relation with congruence lattice representation problems, we consider the problem whether such a measure can be extended to a poset measure μ : P ×P → S, for a larger poset P , such that for al...

We define a new object, called a signed poset, that bears the same relation to the hyperoctahedral group B n (i.e., signed permutations on n letters), as do posets to the symmetric group S n. We then prove hyperoctahedral analogues of the following results: (1) the generating function results from the theory of P-partitions; (2) the fundamental theorem of finite distributive lattices (or Birkho...

Let m ≥ 2 be an integer. We say that a poset P = (X, ) is m-partite if X has a partition X = X1 ∪ · · · ∪Xm such that (1) each Xi forms an antichain in P, and (2) x ≺ y implies x ∈ Xi and y ∈ Xj where i, j ∈ {1, . . . , m} and i < j. If P is m-partite for some m ≥ 2, then we say it is multipartite. – In this article we discuss the order dimension of multipartite posets in general and derive tig...