PUB - Publikationen an der Universität Bielefeld

Given a finite set E of cardinality, say, N and a map f : P(E) --> ($) over bar R := RU{-infinity}, we define a sequence e(1),...,e(N) of elements of E to be f-greedy if e(i) is not an element of {e(1),...,e(i)-1} and f({e(1),...,e(i)}) greater than or equal to f({e(1),...,e(i-1} boolean OR {e}) for all i is an element of {1,...,N} and all e is an element of E\{e(1),...,e(i-1)}. In addition, for any map eta : E --> R we put f(eta)(x) := f(x) + sigma(e is an element of x) eta(e) for all x subset of or equal to E, and we define f to be a well-layered map if for every eta : E --> R, every f(eta)-greedy sequence e(1),...,e(N) is an element of E and every layer p(i)(E) := {y subset of or equal to E parallel to #y = i} (i = 0,..., N) we have f(eta)({e(1),...,e(i)}) = max{f(eta)(y) parallel to y is an element of p(i)(E)}. In this note, we characterize well-layered maps by some appropriate quantified version of the greedoid exchange condition or-quivalently-by some appropriate greedoidal version of the quantitative relations defining valuated matroids and valuated Delta-matroids, respectively.