Keith Peterson wrote:

> Partition implication is defined by David P. Ellerman as follows...

When possible, a very nice way to define implication as an operation

$$ \to : A \times A \to A $$

in a poset \\(A\\) is by demanding that it give a right adjoint to \\( \wedge\\) in the following sense:

$$ p \wedge q \le r \textrm{ if and only if } p \le (q \to r) $$

for all \\(p,q,r \in A\\). Or in words, "\\(p \wedge q\\) implies \\(r\\)" is true if and only if "\\(p\\) implies \\(q \to r \\)".

Here we have to carefully distinguish implication as a binary relation on \\(A\\), which is \\(\le\\), from implication as a binary operation on \\(A\\), which I'm calling \\(\to\\) but is also called \\( \supset \\) or other things.

**However,** a poset with all meets and an operation \\(\to\\) that's right adjoint to \\( \vee \\) in the above sense is "cartesian closed", and in this situation binary meets distribute over joins, which fails in the lattice of partitions. So, in fact, the lattice of partitions does not admit an operation \\(\to\\) as above.

> Partition implication is defined by David P. Ellerman as follows...

When possible, a very nice way to define implication as an operation

$$ \to : A \times A \to A $$

in a poset \\(A\\) is by demanding that it give a right adjoint to \\( \wedge\\) in the following sense:

$$ p \wedge q \le r \textrm{ if and only if } p \le (q \to r) $$

for all \\(p,q,r \in A\\). Or in words, "\\(p \wedge q\\) implies \\(r\\)" is true if and only if "\\(p\\) implies \\(q \to r \\)".

Here we have to carefully distinguish implication as a binary relation on \\(A\\), which is \\(\le\\), from implication as a binary operation on \\(A\\), which I'm calling \\(\to\\) but is also called \\( \supset \\) or other things.

**However,** a poset with all meets and an operation \\(\to\\) that's right adjoint to \\( \vee \\) in the above sense is "cartesian closed", and in this situation binary meets distribute over joins, which fails in the lattice of partitions. So, in fact, the lattice of partitions does not admit an operation \\(\to\\) as above.