Let $\mathfrak{A}$ be a poset, $\mathfrak{B}$ and $\mathfrak{C}$ be meet-semilattices with least elements. Let $f:\mathfrak{A}\rightarrow\mathfrak{B}$ and $g:\mathfrak{A}\rightarrow\mathfrak{C}$ are order embeddings.
Can we warrant that $f(x)\cap^{\mathfrak{B}}f(a) = f(y)\cap^{\mathfrak{B}}f(a) \Leftrightarrow g(x)\cap^{\mathfrak{C}}g(a) = g(y)\cap^{\mathfrak{C}}g(a)$ for every $x,y,a\in\mathfrak{A}$? How to prove this?