\documentclass{PJM}


\authorone{Shalosh B. EKhad and Doron Zeilberger}
\addressone{Department of Mathematics,
Rutgers University (New Brunswick)
Piscataway, NH 08544}
\countryone{U. S. A.}
\emailone{DoronZeil@gmail.com }
\title{TITLE }
\headlinetitle{ Invariance Properties of Matrix Powers}
\begin{document}
\centerline{Communicated  by  Ayman Badawi}
\vskip0.2in
\footnotesize MSC 2010 Classifications: Primary 05A30
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Keywords and phrases: Matrix powers, Larcombe relations, bijective proofs, weight-enumerator
\vskip0.1in
\normalsize

 {\bf Abstract} 


\section{Introduction}


\section{Section 1}

%KAKI
\begin{lemma} \label{lemma2.1}
The following conditions are equivalent: \\
\indent (1) $S$ is a finite semigroup with no proper subsemigroups; \\
\indent (2) $S$ is a trivial group (that is, a group with only one
element).
\end{lemma}

\noindent {\bf Proof.} 

\begin{proposition} \label{prop2.2}
KAKI
\end{proposition}

\noindent {\bf Proof.} 

\begin{corollary} \label{cor2.3}
KAKI
\end{corollary}

\noindent {\bf Proof.} KAKI

\begin{remark} \label{remark2.4}
\hspace*{2mm} 
KAKI
\end{remark}


\begin{thebibliography}{99}
\bibitem{A} K. E. Aubert, On the ideal theory of commutative semi-groups,
{\em Math. Scand.} {\bf 1}, 39--54 (1953).




\end{thebibliography}
\end{document}

