diff --git a/papers/software_fmea/software_fmea.tex b/papers/software_fmea/software_fmea.tex
index 22815dd..125c80d 100644
--- a/papers/software_fmea/software_fmea.tex
+++ b/papers/software_fmea/software_fmea.tex
@@ -65,6 +65,8 @@
 \newcommand{\fm}{failure~mode}
 \newcommand{\fms}{failure~modes}
 \newcommand{\fg}{functional~group}
+\newcommand{\FG}{\mathcal{G}}
+\newcommand{\DC}{\mathcal{DC}}
 \newcommand{\fgs}{functional~groups}
 \newcommand{\dc}{derived~component}
 \newcommand{\dcs}{derived~components}
@@ -345,7 +347,12 @@ of the {\fg} that it was derived from.
 % a {\fg}. Our use of it as a building block corresponds to a {\dc}.
 
 We can use the symbol `$\bowtie$' to represent the creation of a derived component 
-from a {\fg}. We show an FMMD hierarchy in figure~\ref{fig:fmmdh}.
+from a {\fg}. This symbol is convenient for drawn hierarchy diagrams. % (see figure~\ref{fmmdh}). 
+We define the $\bowtie$ function, where $\FG$ is the set of all {\fgs} and $\DC$ is the set of all {\dcs},
+
+$$ \bowtie ( {\FG} ) \mapsto {\DC} .$$
+
+We show an FMMD hierarchy in figure~\ref{fig:fmmdh}.
 Using this diagram, we can follow the creation of the hierarchy in 
 a theoretical system.
 %
@@ -920,9 +927,14 @@ as a hierarchical diagram, see figure~\ref{fig:hd}.
 
 
 
+We can represent the hierarchy in figure~\ref{fig:hd} algebraically, using the $\bowtie$ function
+definition nested thus:
 
+$$ \bowtie \Big( \bowtie \big( \bowtie(R,ADC), read\_4\_20\_input \big), read\_4\_20\_input \Big). $$ 
 
-
+This nested structure means that we have multiple traceable
+stages of failure mode reasoning in our analysis. Traditional FMEA would have only one stage 
+of reasoning for each component failure mode.
 
 %\clearpage
 \section{Conclusion}