For example, in the following diagram I want to allow the rows as in the picture below (i.e, so that m in the bottom row aligns with the top of the fraction it is adjacent to, and so that the top row aligns in the way in the picture)? How can I do this?

\usepackage{pgfpages,amsmath,amssymb,mathtools,color, graphicx, amsfonts, braket}


\begin{matrix} \begin{array}{c|c}
\multicolumn{2}{c}{\Diamond \alpha}
\end{array} \\[3ex]\text{exp} \\[1ex] m \end{matrix}
\end{pmatrix}^{\uparrow} =
\kern15pt& \begin{matrix}
\multicolumn{2}{c}{\cfrac{\ \Diamond \beta\ }{\ \alpha\ }}
 \end{array}   & & \begin{array}{c|c}
\multicolumn{2}{c}{\cfrac{\ \Diamond \beta\ }{\ \alpha\ }}
 \end{array}   \\[3ex]
\text{exp}         & = &  \text{exp}\\[1ex]
 m^{\uparrow} &\kern15pt& \cfrac{\ \operatorname{m \bigstar \lambda v.     [\;]}\ }{\ v\ }

enter image description here

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