As noted by John in #15 above, the implication for partition logic is not defined by the usual adjunction, but it has a cool definition equivalent to the one mentioned by Keith Peterson in #10 where the ditset of a partition is just the complement of the partition as a binary equivalence relation, and the interior operation on any subset of \\(X^2\\) is the complement of the reflexive-symmetric-transitive closure of the complement--which is not a topological interior operation (otherwise partition logic would just be intuitionistic logic). In other words, the ditset of a partition is just the set of ordered pairs from the underlying set that are in different blocks of the partition. Such an ordered pair is called a "distinction" or "dit" of the partition, and the "Yoga" of the logic of partitions relative to logic of subsets is that a dit of a partition is the correlate of an element of a subset.

The cooler but equivalent definition of the implication \\(Q\Rightarrow P\\) is that it is partition like \\(P\\) except that whenever a part or block \\(B\\) of \\(P\\) is contained in a block of \\(Q\\), then that block of \\(P\\) is replaced by the discretized version, i.e., all the singletons of the elements in \\(P\\). When the lattice of partitions is presented with the refinement partial order, then the top is the discrete partition of all singletons. The relation of the implication operation to the partial order is the same as in a Boolean algebra of subsets, namely if the implication is equal to the top, then the partial order holds between those two partitions (or subsets in the BA case).

The full definition of all the binary 'Boolean' operations on partitions and the correctness and completeness theorems for a system of semantic tableaus for partition logic are all given in the paper on the logic of partitions cited in my Introduction.

The cooler but equivalent definition of the implication \\(Q\Rightarrow P\\) is that it is partition like \\(P\\) except that whenever a part or block \\(B\\) of \\(P\\) is contained in a block of \\(Q\\), then that block of \\(P\\) is replaced by the discretized version, i.e., all the singletons of the elements in \\(P\\). When the lattice of partitions is presented with the refinement partial order, then the top is the discrete partition of all singletons. The relation of the implication operation to the partial order is the same as in a Boolean algebra of subsets, namely if the implication is equal to the top, then the partial order holds between those two partitions (or subsets in the BA case).

The full definition of all the binary 'Boolean' operations on partitions and the correctness and completeness theorems for a system of semantic tableaus for partition logic are all given in the paper on the logic of partitions cited in my Introduction.