Monotone sets and growth in the Hypercube

1. Is the known definition for a monotone set in the hyperplane is : a set $A \in${$0,1$}$^n$ such that for every $a \in A$ if $b < a$ so $b \in A$?

2. Given all hamming balls and all boxes (vector subspaces) in {$0,1$}$^n$. Can I construct a genereal set $A$ in the hypercube, as a minkowski sum of balls and boxes, that is defined uniquely?

3. What is known about the sequence $|A|,|A+B|,|A+2B|,\ldots$ for A a general set in hypercube, and B is the hamming ball of radius 1?

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1) Yes; 2) No - consider the set {(1,1,1,1,1)} it's not the Minkowski sum of anything; 3) If $A$ is a singleton, the cardinalities are known. Similarly if $A$ is $2k$-discrete, then the $|A+iB|$ are known for $i\le k$. –  Anthony Quas Dec 14 '12 at 17:09
Can you please write the definitions of $2k$-discrete sets and singleton sets (or reference me to these definitions)? thanks –  ak47 Dec 15 '12 at 15:04
ok, got you. it's quite trivial :) –  ak47 Dec 15 '12 at 17:02