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Let $Alg$ be a single-sorted finitary variety i.e. one specified by a set $\Sigma$ of operations with finite arity and also a set $E$ of equations, the morphisms being the usual $\Sigma$-algebra morphisms.

Further let $T : Alg \to Alg$ be a finitary functor that preserves monos (injections) and empty binary intersections i.e. $T(A \cap B) = TA \cap TB$ whenever $A \cap B = \emptyset$. This trivially holds if $\Sigma$ contains a constant.

Then my question is:

Does there exist such a $T$ that fails to preserve finite or infinite intersections?

Note that every $T : Set \to Set$ preserves non-empty finite intersections by page 3 of:

so under my assumptions $T$ would preserve finite intersections. Then if $T$ is finitary this can be shown to imply preservation of infinite intersections. Therefore no such example can be found in $Set$, this being the variety where $\Sigma = E = \emptyset$.

Any help much appreciated.

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The endofunctor T of Set sending a set X to a singleton 1 if it is non-empty, and to the empty set 0 if it is empty preserves monomorphisms, since it sends every map to a monomorphism. But it does not preserve the intersection of the two maps from a singleton 1 to a two-element set 2. – Steve Lack Jan 9 '12 at 23:03
@Steve: Thanks. However I was intending to exclude this case because one can alter the functor so that it does preserve finite intersections, whilst preserving the structure of $T$-algebras and $T$-coalgebras. I have edited the question accordingly. – Rob Myers Jan 10 '12 at 0:05
Rob, I'm not sure quite what this means. I take it that the modified version sends everything to 1? In that case the empty set will no longer possess an algebra structure. – Steve Lack Jan 10 '12 at 2:15
Ok, you are again correct. In "Terminal coalgebras in well-founded set theory" Barr shows one can alter $T : Set \to Set$ on $\emptyset$ to ensure it preserves monos, so that the $T$-algebras and $T'$-algebras are isomorphic. However he assumes $T\emptyset \neq \emptyset$, which excludes your example. A similar construction for finite interesection preservation is given here: (page 4), which explicitly mentions your example. Really I am looking to exclude such cases i.e. failure to preserve empty intersections. – Rob Myers Jan 10 '12 at 9:32
An empty intersection of subobjects is the whole object, so is always preserved. The example I gave involved failure to preserve a binary intersection. Perhaps it would be helpful if you explained what you were really after. – Steve Lack Jan 10 '12 at 9:44

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