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\section*{co-Heyting algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


[[!redirects coHeyting algebra]] [[!redirects Co-Heyting algebra]] [[!redirects CoHeyting algebra]] [[!redirects bi-Heyting algebra]] [[!redirects biHeyting agebra]] [[!redirects Brouwerian algebra]]

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

When the [[lattice of open subsets]] of a [[topological space]] is the primordial example of a [[Heyting algebra]] then its dual \emph{lattice of [[closed subsets]]} is the primordial example of a \textbf{co-Heyting algebra}.

In general, co-Heyting algebras are [[duality|dual]] to Heyting algebras and like them come equipped with non-Boolean logical operators that make them interesting players in [[modal logic|modal]], [[paraconsistent logic|paraconsistent]], and co-intuitionistic logic, [[linguistics]], [[topos theory]], [[continuum physics]], [[quantum theory]] and in [[mereology]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{co-Heyting algebra} is a bounded [[distributive lattice]] $L$ equipped with a binary \emph{[[subtraction]]} operation $\backslash :L\times L\to L$ such that $x\backslash y\leq z$ iff $x\leq y\vee z$.\footnote{Existence of $\backslash$ amounts to an [[adjunction]] \_$\backslash y\dashv y\vee$\_ and the existence of a left adjoint implies that $y\vee$\_ preserves limits $\wedge$ hence the assumption of distributivity in the definition is redundant and has been put in for emphasis only.} 

A \textbf{bi-Heyting algebra} is a bounded distributive lattice $L$ that carries a Heyting algebra structure with implication $\Rightarrow$ and a co-Heyting algebra structure with subtraction $\backslash$.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Co-Heyting algebras were initially called \emph{Brouwerian algebras} . Bi-Heyting algebras were introduced and studied in a series of papers by Cecylia Rauszer in the 1970s who called them \emph{semi-Boolean algebras} which suggests to view them as a generalization of [[Boolean algebra|Boolean algebras]].


\item A [[topos]] $\mathcal{E}$ such that the lattice $sub(X)$ of subobjects is a bi-Heyting algebra for every object $X\in\mathcal{E}$ is called a [[bi-Heyting topos]].



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The lattice of closed subsets of a topological space is a co-Heyting algebra with $X\backslash Y=\overline{X\cap Y^c}\quad$.


\item A Boolean algebra provides a (degenerate) example of a bi-Heyting algebra by setting $x\Rightarrow y:=\neg x\vee y$ and $x\backslash y:=x\wedge\neg y$.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item $a\backslash b=0$ iff $a\backslash b\leq 0$ iff $a\leq b\vee 0$ iff $a\leq b$. In particular, $a\backslash a=0$.


\item As \_$\backslash x$ has a right adjoint it preserves colimits hence: $(a\vee b)\backslash x =(a\backslash x)\vee (b\backslash x)\quad$.


\item $a\backslash 0\leq a\backslash 0$ iff $a\leq 0\vee (a\backslash 0)$ iff $a\leq a\backslash 0\quad$. On the other hand, $a\leq 0\vee a$ and the adjunction yield $a\backslash 0\leq a\quad$, hence $a\backslash 0 = a\quad$.


\item Suppose $a\leq b\vee x$ then $a\backslash b\leq x$. As from $a\backslash b\leq a\backslash b$ follows $a\leq b\vee (a\backslash b)$, hence $a\backslash b =\Wedge\{x|a\leq b\vee x\}\quad$.


\item The subtraction operation permits to define the [[co-Heyting negation]] $\sim: L\to L$, called \emph{non a} in Lawvere (1991), by setting $\sim a:=1\backslash a$.


\item $\sim$ in turn can then be used to define the [[co-Heyting boundary|co-Heyting boundary operator]] $\partial :L\to L$ by $\partial a:=a\wedge\sim a$. That $\partial a$ is not necessary trivial is dual to the non-validity of the \emph{tertium non datur} for general Heyting algebras and points to the utility of co-Heyting algebras for [[paraconsistent logic]].


\item In [[bi-Heyting toposes]] like e.g. [[essential subtoposes]] of presheaf toposes (\hyperlink{Law91a}{Lawvere}, \hyperlink{Reyes}{Reyes}), the co-Heyting algebra operations are generally not preserved by [[inverse image functor|inverse image functors]], so that the co-Heyting logical operators are subject to \emph{[[de re and de dicto]]} effects. The parallel between this and the [[commutator]] in quantum mechanics has been suggested by Lawvere thereby somewhat anticipating the view of D\"o{}ring (2013).



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[co-Heyting negation]]
\item [[co-Heyting boundary]]
\item [[Heyting algebra]]
\item [[Heyting category]]
\item [[distributive category]]
\item [[mereology]]
\item [[bi-Heyting topos]]
\item [[bitopological space]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item G. Bellin, \emph{Categorical Proof Theory of Co-Intuitionistic Linear Logic} , arXiv:1407.341 (2014). (\href{http://arxiv.org/pdf/1407.3416v1}{pdf})


\item G. Bezhanishvili, N. Beshanishvili, D. Gelaia, A. Kurz, \emph{Bitopological duality for distributive lattices and Heyting algebras} , Math. Struc. Comp. Sci. \textbf{20} no.3 (2010) pp.359-393. (\href{http://www.cs.le.ac.uk/people/nb118/Publications/PairwiseStone.pdf}{preprint})


\item [[Andreas Döring|A. Döring]], \emph{Topos-based Logic for Quantum Systems and Bi-Heyting Algebras} , arXiv:1202.2750 (2013). (\href{http://arxiv.org/pdf/1202.2750.pdf}{pdf})


\item M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{The non-Boolean logic of natural language negation} , Phil. Math. \textbf{2} no.1 (1994) pp.45-68.


\item M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Models for non-Boolean negation in natural languages based on aspect analysis} , pp.241-260 in Gabbay, Wansing (eds.), \emph{What is Negation?}, Kluwer Dordrecht 1999.


\item M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Generic Figures and their Glueings} , Polimetrica Milano 2004.


\item [[William Lawvere|F. W. Lawvere]], \emph{Introduction} , pp.1-16 in Lawvere, Schanuel (eds.), \emph{Categories in Continuum Physics} , Springer LNM \textbf{1174} 1986.


\item [[William Lawvere|F. W. Lawvere]], \emph{Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes} , pp.279-281 in A. Carboni, M. Pedicchio, G. Rosolini (eds.) , \emph{[[Como|Category Theory - Proceedings of the International Conference held in Como 1990]]} , LNM \textbf{1488} Springer Heidelberg 1991.


\item T. Mormann, \emph{Heyting Mereology as a Framework for Spatial Reasoning} , Axiomathes \textbf{23} no.1 (2013) pp.237-264. (\href{http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf}{draft})


\item [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{Bi-Heyting Algebras, Toposes and Modalities} , J. Phi. Logic \textbf{25} (1996) pp.25-43.


\item C. Rauszer, \emph{Semi-Boolean algebras and their applications to intuitionistic logic with dual operations} , Fund. Math. \textbf{83} no.3 (1974) pp.219-249. (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83120.pdf}{pdf})


\item J.G. Stell, M.F. Worboys, \emph{The algebraic structure of sets of regions} , pp.163-174 in Hirtle, Frank (eds.), \emph{Spatial Information Theory}, Springer LNCS \textbf{1329} (1997).


\item [[Marshall Stone|M. H. Stone]], \emph{Topological representation of distributive lattices and Brouwerian logics} , Cas. Mat. Fys. \textbf{67} (1937) pp.1-25. (\href{http://dml.cz/bitstream/handle/10338.dmlcz/124080/CasPestMatFys_067-1938-1_1.pdf}{pdf})


\item F. Wolter, \emph{On logics with coimplication} , J. Phil. Logic \textbf{27} (1998) pp.353-387. (\href{http://lips.informatik.uni-leipzig.de/files/1998-24.pdf}{draft})



\end{itemize}


\end{document}