construct a bijective map between subsets of binary sequence

Question

Consider the binary sequence $\{0,1\}^N$ where $N$ is an even integer (for simplicity). Let $M_k := \{\beta\in \{0,1\}^N \rvert \sum_{j=1}^N \beta_j = k\}$ (i.e., $M_k$ is the set that contains all binary sequences with length $N$ and with exactly $k$ 1's). The question is: for general $k \le N/2$, is it always possible to construct a bijective map $\phi: M_k\rightarrow M_{N-k}$ such that $\phi(\beta)_j \ge \beta_j$? (Here, $\phi(\beta)_j$ is the $j^{th}$ coordinate of the binary sequence $\phi(\beta)$.)

Obviously, both sets $M_k$ and $M_{N-k}$ have the same number of elements. When $k = 1$, the construction of $\phi$ is simple. However, for general $k$ and $N$, I am not sure how to systematically construct such a function $\phi$. Any hint or reference would also help a lot. Thank you.

Answer 1

The answer is "yes, this is always possible." The bijection on ranks $k$ and $N-k$ induced from a symmetric chain decomposition of the Boolean lattice gives you what you're asking for. For basics on this topic, see these slides.