 Definition (Ploščica, Tůma, and Wehrung 1998).Let "e" be an element in a (∨,0)-semilattice "S".We say that the "weak uniform refinement property" WURP holds at "e", if for all families formula_12 and formula_13 of elements in "S" such that "ai ∨ bi=e" for all "i" in "I", there exists a family formula_14 of elements of "S" such that the relations

• "ci,j ≤ ai,bj",

• "ci,j ∨ aj ∨ bi=e",

• "ci,k ≤ ci,j∨ cj,k"

hold for all "i, j, k" in "I".