## Lists

Lists are a very common structure in mathematical papers. For example, a list may represent different parts of a theorem, different parts of a proof, a list of properties, etc. Latex has several pre-defined environments (itemize, enumerate, description) for typesetting lists. Alternatively, a list can also be typeset without any special coding other than inserting paragraph breaks (i.e., one or more blank lines) between the list items, or inline, with commas or semicolons separating the list items.

The main considerations in typesetting a list are (i) the choice of environment (if any) for the list; (ii) the style of labels; and (iii) the formatting (if any) of the labels.

My advice on these issues is the following:

### Use labels with balanced parentheses, e.g., "(i)", "(a)", "(P)".

Labels with unbalanced parentheses, such as "1)", look ugly, and they can mess with the parenthesis matching of an editor. Labels with a period at the end (e.g. "1.") make references to these labels tricky. For example, "By 1. we have" doesn't look good, but the alternative "By 1 we have" has its own problem. Using labels with balanced parentheses avoids these problems.

### Use lower case Roman letters (e.g., "(i)"), rather than numerals, as labels for parts of a theorem

This is the most common practice in mathematical writing. Its main advantage is that it avoids conflicts with numbered equations. Even if one doesn't have numbered equations in a paper, using numbered labels still may cause confusion, since upon seeing a reference like "by (1) we have ..." most readers would instinctively assume that (1) refers to an equation and hunt for an equation numbered (1).

### Avoid using explicit fonts or special formatting in conjunction with labels

I often see papers in which the authors try to achieve a particular "look" of a label, by placing the label, or a reference to it, inside a pair of dollar signs ("$(i)$"), or within a font instruction ("\textit{(i)}"). This is unnecessary, and it is contrary to standard typesetting practice. Moreover, the dollar sign version violates the basic rule of using math mode (i.e., $...$) only for mathematical material.

(There is one situation in which explicit formatting may be appropriate: In a list inside theorem, it may be appropriate to enclose "(i)" in a "\textup{...}" to prevent it from being italicized; i.e., instead of \item[(i)] use \item[\textup{(i)}].)

### Use the itemize or enumerate environment for lists in theorems, or other lists with short items (no more than a few lines each). Use no special environment, other than paragraph breaks separating the items, for lists in proofs, or "long" lists

The list environments cause the entire list to be indented. This is the desired look for a short list (e.g., in a theorem, or a list of properties), but it is inappropriate for longer lists, especially if they extend over more than one page.

To generate the item labels, put the desired label in brackets after "\item", e.g., "\item[(i)]", etc.

If explicit labels are given, there is no essential difference between the itemize and enumerate environments, so either one can be used. Without such explicit labels, an enumerate environment generates numbered labels (though usually not in the desired style), whereas an itemize environment produces a bullet list.

### Lists at the beginning of a theorem statement. The \mbox{} trick.

Anyone who has had to typeset theorems with lists probably has run into the problem that if the list occurs at the beginning of the statement (i.e., with no text preceding it inside the theorem), it gets typeset right after the theorem name, and not on a new line and indented in the same way as the other items, and putting spacing commands or explicit newlines before the list doesn't seem to help. The reason for this behavior is that the theorem environment "eats up" (nearly) any space at the beginning of the theorem, so commands like \hfill or \newline or \par have no effect. The remedy is to trick TeX into thinking that there is material before the list, by putting an "\mbox{}" before the list. See the example below.

### An example. A three part theorem, with proof

Here is an example that illustrates a typical situation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\mbox{}
\begin{itemize}

\item[(i)] $QP\cap \Omega \subset Q$.

\item[(ii)] If $(X,T)$ is minimal, then $QP=PQ$.

\item[(iii)]  Let $\pi:X \to Y$ be a homomorphism of minimal
flows. Then $\pi(Q_XP_X)=Q_YP_Y$.

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{proof}

(i) Let $(x,z)\in QP\cap \Omega$. Let $y\in X$ such that
$(x,y)\in Q$ and $(y,z)\in P$. Let $I$ be a minimal right ideal
in $E(X)$ such that $yr=zr$ for all $r\in I$, and let $u$ be an
idempotent in $I$ such that $(x,z)u=(x,z)$. Then $(x,z)=(x,z)u= (x,y)u \in Qu\subset Q$.

(ii)  Note that if the relation $R$ is symmetric, so is
$C(R)$. Therefore $C(Q)$ is symmetric. Now $C(Q)=PQ$ (\cite{AGl}), so
$QP=(PQ) ^{-1}=C(Q)^{-1}=C(Q)=PQ$.

(iii)  Let $(y_1,y_2)\in Q_Y$ and $(y_2,y_3)\in P_Y$. Let
$v$ be a minimal idempotent in $E(Y)$ such that $y_3=y_2v$, and
let $w$ be a minimal idempotent in $E(X)$ such that $\theta (w)=v$. Let $(x_1,x_2)\in Q(X)$ such that $\pi(x_1,x_2)= (y_1,y_2)$. Then $\pi(x_1,x_2w)=(y_1,y_2v)=(y_1,y_3)$ and $(x_1, x_2w)\in Q_XP_X$.
\end{proof}