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\def\twoB{\bar 2} %mh
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\def\ISP{\mathbb{ISP}\,}
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\begin{document}

\title{An efficient transfer of strong dualities}
\author{Brian A. Davey and Miroslav Haviar$^*$}
\date{}
\maketitle

\noindent A \defn{full duality} for the quasi-variety $\CA :=
\ISP(\DB)$ generated by a finite algebra $\DB = \langle D;
F\rangle$ is a dual category equivalence between $\CA$ and the
category $\CX := \IScP(\DT)$ where $\DT = \langle D; G, H, R,
\T\rangle$ is a (discrete) topological structure allowing sets
$R$ of relations, $G$ of operations and $H$ of partial operations
in its type.  Here $\mathbb I$, $\mathbb S$ $\mathbb P$
(arbitrary index sets) and $\mathbb P^+$ (non-empty index sets),
denote the usual class operators while $\mathbb S_c$ denotes
``{\em closed substructure of\/}".  The duality is called
\defn{strong} if it is full and moreover $\MT$ is injective
in~$\CX$.  Strong dualities provide a powerful tool for the study
of the quasi-variety~$\CA$.  Since a quasi-variety may have many
different generating algebras it is useful to know how to take a
known strong duality for $\CA$ based on a finite generating
algebra $\DB$ and transfer it to a strong duality based on another
finite generating algebra~$\MB$.  Our concern here is not the {\em
existence} of a strong duality based on $\MB$: that is known---see
Davey and Willard~\cite{DaveyWillard:Indep} and
Saramago~\cite{Saramago:DandEndoD} for the existence of a duality
and Hyndman~\cite{Hyndman:StrongIndep} for the existence of a
strong duality. The aim of this paper is to give conditions under
which we obtain a particularly efficient and user friendly
transfer of a strong duality for $\CA$ based on $\DB$ to a strong
duality based on~$\MB$.

We show that, in many cases, if $\DB$ and $\MB$ are finite
algebras which generate the same quasi-variety $\CCD$, then a
strong duality for $\CCD$ based on $\DB$ may be transferred to a
strong duality for $\CCD$ based on $\MB$ by the addition of some
endomorphisms of $\MB$ and just one further partial operation.
This additional operation exhibits the schizophrenia so typical of
the theory of natural dualities. We show how the result may be
applied to yield an efficient strong duality in the case when $\MB$
is a distributive lattice, a semilattice or an abelian group.

\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{1}

\bibitem{quest}
D.M. Clark and B.A. Davey, \emph{The quest for strong dualities}, J. Austral.
  Math. Soc. Ser. A \textbf{58} (1995), 248--280.

\bibitem{CD98}
D.M. Clark and B.A. Davey, \emph{{Natural Dualities for the Working
  Algebraist}}, Cambridge University Press, Cambridge, 1998.

\bibitem{Houston}
B.A. Davey and M.~Haviar, \emph{A schizophrenic operation which aids the efficient
transfer of strong dualities}, submitted.

\bibitem{DaveyWillard:Indep}
B.A. Davey and R.~Willard, \emph{The dualisability of a quasi-variety is
  independent of the generating algebra}, submitted.

\bibitem{Hyndman:StrongIndep}
J.~Hyndman, \emph{Strong duality of finite algebras that generate the same
  quasivariety}, preprint.

\bibitem{Saramago:DandEndoD}
M.J. Saramago, \emph{Some remarks on dualisability and endodualisability},
  submitted.

\end{thebibliography}

%\end{document}

\noindent Department of Mathematics, La Trobe University, Bundoora 3083, Victoria,
Australia

\noindent \texttt{B.Davey@latrobe.edu.au,}
\noindent \texttt{M.Haviar@latrobe.edu.au}


\end{document}