A Jbre F of a partially ordered set P is a subset which intersects each nontrivial maximal antichain of P. Let I be the smallest constant such that each finite partially ordered set P contains a fibre of size at most I ‘1 PI. We show that 1, < 3 by finding a good 3-coloring of the nontrivial antichains of P. Some decision problems involving iibres are also considered. For instance, it is shown ...

Set-valued ordered information systems can be classified into two categories: disjunctive and conjunctive systems. Through introducing two new dominance relations to set-valued information systems, we first introduce the conjunctive/disjunctive set-valued ordered information systems, and develop an approach to queuing problems for objects in presence of multiple attributes and criteria. Then, w...

Definition. A specific set that we often care about is the empty set, ∅ i.e. the set containing no elements. One particular quirk of the empty set is that any statement of the form ∀x ∈ ∅ . . . will always be vacuously true, as it is impossible to disprove (as we disprove ∀ claims by using ∃ quantifiers!) For example, the statement “every element of the empty set is delicious” is true. Dumb, bu...

Given an integer n 2, let H n be the set H n = {(a, b) : ab ≡ 1 (mod n), 1 a, b n − 1} and let M (n) be the maximal difference of b − a for (a, b) ∈ H n. We prove that for almost all n, n − M (n) = O n 1/2+o(1). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n .