
\subsection{Full \dpg} \label{ssec:FullDPG}

The full \dpg\ category needed in this paper is merely a ``garnished''
version of $\dpg_b$, in which every variable has a ``weight'', and some
weight restriction apply. We now turn to its formal definition, which
we give in a slightly informal manner.

\begin{context}
Let $n>0$ be a positive integer, and let us work in some universe of
Latin and Greek variables in which every variable $z$ (or $\zeta$) has a
weight $\wt(z)$ (or $\wt(\zeta)$) with $0\leq\wt(z),\wt(\zeta)\leq n$,
so that if $z$ and $\zeta$ are dual then $\wt(z)+\wt(\zeta)=n$. Every
monomial in our universe now has a weight, the sum of the weights of all
the variables appearing in it, counted with multiplicity. The variables
$\hbar$ and $\eps$ are special and do not carry a weight.
\end{context}

\begin{example} In the main context of this paper, that of
Section~\ref{sec:Everything}, we will have variables $y_i$,
$b_i$, $a_i$, and $x_i$ (where $i$ can run in some sets of
labels), and their duals $\eta_i$, $\beta_i$, $\alpha_i$,
and $\xi_i$, with weights $\wt(y_i,b_i,a_i,x_i)=(1,0,2,1)$ and
$\wt(\eta_i,\beta_i,\alpha_i,\xi_i)=(1,2,0,1)$. In this context,
$\wt(\alpha_3^{62}a_1^8y_{41}^3\hbar^1\eps^7)=62\cdot 0+8\cdot 2+3\cdot 1+1\cdot 0+7\cdot 0=19$.
\end{example}

\begin{definition} A power series $D=\sum D^{(k)}\eps^k$ is called ``docile''
if for every $k$ every monomial appearing in $D^{(k)}$ has weight less than
$n(k+1)$ (with a slight imprecision, this is $\wt(D^{(k)})\leq n(k+1)$). The
same $D$ is called ``$G$-docile'' if it is docile and in addition the following
``Condition $G_{n0}$'' holds:
\vskip 2mm
\begin{quote}
{\bf Condition $G_{n0}$.} For any weight-$n$ variable $z$, $\partial_zD^{(0)}$
is affine-linear in the weight-$0$ variables.
\end{quote}
\end{definition}

\begin{comment} Note that if $D$ is docile then $\wt(D^{(0)})\leq n$ so if
also $wt(z)=n$, then $\wt(\partial_zD^{(0)})=0$. 
\end{comment}

{\red Possible improvement: \dpg\ are things which are $\eps$-weight-docile, $\hbar$-Latin-docile, and have no
Greek-only pairs. Can the last condition also be phrased as a docility condition?}

{\red MORE: State up front a full EDDO/\dpg\ theorem.}

The diagrammatic discussion of this section can be continued and
extended to the full $\dpg_n$ category of Section~\ref{ssec:FullDPG}
but we prefer the more solid grounds of pure algebra as in the next
section, Section~\ref{ssec:PDE}.

{\red MORE.}
