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An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in Rn. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application,...
We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying this condition, we propose polynomial algorithms to solve the jump number problem.
Posets with property DINT which are Priestely spaces with respect interval topologies are characterized. Also, posets which are Priestley spaces with respect to bi-Scott topologies are characterized.
The sizes of Boolean combinations of subgroups Gi of a finite abelian group depends only on the Boolean expression, the 0-1-sublattice generated by the Gi, and the size of minimal subquotients from this sublattice. Moreover, they increase, monotonically, with those sizes.
Garrett Birkhoff conjectured in 1942 that when A, B, P are finite posets satisfying AP≅BP, then A≅B. We show that this is true. Further, we introduce an operation C(AB), related to Garrett Birkhoff's exponentiation, and determine the structure of the algebra of isomorphism types of finite posets under the operations induced by A+B, A×B, and C(AB). Every finite +-indecomposable and ×-indecomposable...
Lacking an explicit formula for the numbers T0(n) of all order relations (equivalently: T0 topologies) on n elements, those numbers have been explored only up to n=13 (unlabeled posets) and n=15 (labeled posets), respectively. In a new approach, we used an orderly algorithm to (i) generate each unlabeled poset on up to 14 elements and (ii) collect enough information about the posets on 13 elements...
A topological space X is said to be splittable over a class P of spaces if for every A⊂X there exists continuous f:X→Y∈P such that f(A)∩f(X∖A) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets...
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