‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

Dragan Mašulović Abstract. One can turn the set of retractions of a lattice 〈L,≤〉 into a poset Rf (L) by letting f ≤ g iff f(x) ≤ g(x) for all x ∈ L. In 1982 H. Crapo raised the following two problems: (1) Is it true that Rf (L) is a lattice for any lattice L? (2) Is it true that Rf (L) is a complete lattice if L is a complete lattice? In 1990 and 1991 B. Li published two papers dealing with th...

This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological ...

In this paper X denotes a set. Let L be a lattice. Note that Poset(L) has l.u.b.’s and g.l.b.’s. Let L be an upper-bounded lattice. One can verify that Poset(L) is upper-bounded. Let L be a lower-bounded lattice. One can verify that Poset(L) is lower-bounded. Let L be a complete lattice. Note that Poset(L) is complete. Let X be a set. Then X is an order in X . Let X be a set. The functor 〈X ,⊆〉...

In this article we explore a non-abelian torsion theory in the category of preordered groups: objects its torsion-free subcategory are partially ordered groups, whereas groups (with total order). The reflector from to has stable units, and prove that it induces monotone-light factorization system. We describe coverings relative Galois structure naturally associated with reflector, explain how t...

We investigate whether the completeness of a complete orthonormal sequence for L2(IT, 77) is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence i(277)-1/2emxi we show that given any xb £ L°°(-77, 77), except lb = 0 a.e., there is a nontrivial portion of Í (2rr)-^ einx\ which will maintain completeness under this perturbation. W...