The linear discrepancy of a poset P is the least k for which there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x) − hL(y)| ≤ k, where hL(x) is the height of x in L. In this paper, we consider linear discrepancy in an online setting and devise an online algorithm that constructs a linear extension L of a poset P so that |hL(x) − hL(y)| ≤ 3k − 1, when the lin...

Van der Waerden’s arithmetic sequence theorem in particular, the ‘density version’ of Szemeredi is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and E > 0. Then for every sufficiently large partially ordered set P of width at most u’, every subset S of P satisfying ] S] > E 1 PI contains a chain a, , a2, . . . . a, such that the cardinalit...

According to Larsen [15, p. 13], a function F from a nonvoid subset DF of a commutative semigroup A into A is called a partial multiplier on A if F (D) · E = F (E) ·D for all D, E ∈ DF . Note that if in particular DF is a subsemigroup of A and F ( D · E ) = F (D) · E for all D , E ∈ DF , then F is a partial multiplier on A . A partial multiplier F on A is called total if DF = A . Clearly, the i...

We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already NP-hard on a certain fixed poset, and investigate structural properties of posets yielding fixedparameter tract...

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to...

A general theorem on asymptotically optimal sequential selection of experiments is presented and applied to a Bayesian classification problem when the parameter space is a finite partially ordered set. The main results include establishing conditions under which the posterior probability of the true state converges to one almost surely, and determining optimal rates of convergence. Properties o...