I need the help of a frame theorist in the following problem: Under what assumptions is a dense morphism of frames f:A -> B , actually one to one? I define such a map f to be dense, just in case f(a) = 0 implies a = 0. (Maybe this is the start of my trouble). Then an argument occurs to me that I know must be incorrect, but I just can't get out of my mind: (In what follows, let < stand of less than or equal, and for and element x of a frame A let -x be the largest element of A which is disjoint from x.) Claim: If A is regular and f:A -> B is dense, then it is one to one. The assumptions seems to imply that x is well below y in A iff f(x) is well below f(y) in B. Then, since A is regular, two elements are different just in case one has something well below it that is not well below the other. Can someone lend a hand?
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