Embedding partially ordered sets into chain-products is already known to be NP-complete (see Yannakakis 30] for dimension or Stahl and Wille 26] for 2-dimension). In this paper, we introduce a new dimension parameter and show that encoding using terms (or k-dimension) is not better than bit-vector (or 2-dimension) and vice versa. A decomposition theory is introduced using coatomic lattices. An ...

The most comon definition of the Poset category is incorrect. Here we present a valid categorical concept using morphisms that are “closed” and “complete”.

For an arbitrary partially ordered set P its dual P ∗ is built as the collection of all monotone mappings P → 2 where 2 = {0, 1} with 0 < 1. The set of mappings P ∗ is proved to be a complete lattice with respect to the pointwise partial order. The second dual P ∗∗ is built as the collection of all morphisms of complete lattices P ∗ → 2 preserving universal bounds. Then it is proved that the pa...

The dimension of a partiruly ordered set (X, P) is the smallest positive integer i for which there exists a function f which assigns to each x E X a sequence {f(x)(i): I s i 41) of TeaI numbers so that x Sy in P i! and only if f(x)(i)Sf(y)(i) for each i = 1,2,. , , , t. TIE irltelval ~lmension of (X, P) is the sm;.llcst integer t for which there exists a function F which assigns to 2ach x E X a...

A partially ordered set (X, is called sharply transitive if its automorphism group is sharply transitive on X, that is, it is transitive and the stabilizer of every element is triviaL It is shown that every free group is the automorphism group of a sharply transitive partially ordered set. It is also shown that there exists a sharply transitive partially ordered set (-,Y, :::;) having some maxi...

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...