There are two opposite ways to present what is called the "lattice of partitions".
The older way, e.g., used in Birkhoff's Lattice Theory book and used by Fong and Spivak, is to use "coarser than" partial order on partitions which is just the inclusion relation between the corresponding binary equivalence relations. In the partial order, the indiscrete partition (only one block or part), nicknamed "the blob," is on top since it is the largest equivalence relation, and the discrete partition (all singletons) is on the bottom.
The new way, used in the [logic of partitions](http://www.ellerman.org/introduction-to-partition-logic/), is to use the opposite refinement partial ordering which is the inclusion ordering on the complements of equivalence relations which are called "apartness relations" or "partition relations". These partition relations are characterized as being anti-reflexive, symmetric, and anti-transitive. Then the blob is on the bottom and the discrete partition is on the top.
It is a matter of habit, taste, and the purpose at hand as to which ordering seems more natural. The two lattice operations of join and meet interchange when the ordering is reversed; joins become meets and vice-versa. For instance, the operation defined as the join of partitions in Fong & Spivak is the meet operation in partition logic.
All the usual Boolean operations on subsets can be carried over to partitions so that one can then take an given formula in subset (i.e., 'propositional' logic) and give the variables and operations either a subset interpretation or a partition interpretation. The 'key' to developing partition logic was to define the implication operation on partitions, and then that leads to the notion of negation in partition logic.
Given two partitions \\(P\\) and \\(Q\\), the implication
\\(Q \Rightarrow P\\)
is the partition that is just like \\(P\\) except that whenever a part or block \\(S\\) of \\(P\\) is contained in a part of \\(Q\\), then that block of \\(P\\) is 'discretized', i.e., is replaced by all the singletons of the elements of \\(S\\). Then, as in intuitionistic logic, the negation is the implication to the bottom or zero which is this case is the blob. That blob-negation is trivial since the blob-negation of every partition except the blob is the blob, and the blob-negation of the blob is the top or discrete partition. Of much more interest is the set of all partitions that are implications to a fixed partition \\(P\\), the P-negated partitions, so \\(Q \Rightarrow P\\) is the P-negation of \\(Q\\). They form a Boolean algebra under the partition operations! Thus every partition \\(P\\) determines a Boolean subalgebra of the partition algebra. The P-negation of a P-negated partition just flips the parts \\(S\\) of \\(P\\) between the discretized version (like a mini-one) and the normal version (like a mini-zero).
The older way, e.g., used in Birkhoff's Lattice Theory book and used by Fong and Spivak, is to use "coarser than" partial order on partitions which is just the inclusion relation between the corresponding binary equivalence relations. In the partial order, the indiscrete partition (only one block or part), nicknamed "the blob," is on top since it is the largest equivalence relation, and the discrete partition (all singletons) is on the bottom.
The new way, used in the [logic of partitions](http://www.ellerman.org/introduction-to-partition-logic/), is to use the opposite refinement partial ordering which is the inclusion ordering on the complements of equivalence relations which are called "apartness relations" or "partition relations". These partition relations are characterized as being anti-reflexive, symmetric, and anti-transitive. Then the blob is on the bottom and the discrete partition is on the top.
It is a matter of habit, taste, and the purpose at hand as to which ordering seems more natural. The two lattice operations of join and meet interchange when the ordering is reversed; joins become meets and vice-versa. For instance, the operation defined as the join of partitions in Fong & Spivak is the meet operation in partition logic.
All the usual Boolean operations on subsets can be carried over to partitions so that one can then take an given formula in subset (i.e., 'propositional' logic) and give the variables and operations either a subset interpretation or a partition interpretation. The 'key' to developing partition logic was to define the implication operation on partitions, and then that leads to the notion of negation in partition logic.
Given two partitions \\(P\\) and \\(Q\\), the implication
\\(Q \Rightarrow P\\)
is the partition that is just like \\(P\\) except that whenever a part or block \\(S\\) of \\(P\\) is contained in a part of \\(Q\\), then that block of \\(P\\) is 'discretized', i.e., is replaced by all the singletons of the elements of \\(S\\). Then, as in intuitionistic logic, the negation is the implication to the bottom or zero which is this case is the blob. That blob-negation is trivial since the blob-negation of every partition except the blob is the blob, and the blob-negation of the blob is the top or discrete partition. Of much more interest is the set of all partitions that are implications to a fixed partition \\(P\\), the P-negated partitions, so \\(Q \Rightarrow P\\) is the P-negation of \\(Q\\). They form a Boolean algebra under the partition operations! Thus every partition \\(P\\) determines a Boolean subalgebra of the partition algebra. The P-negation of a P-negated partition just flips the parts \\(S\\) of \\(P\\) between the discretized version (like a mini-one) and the normal version (like a mini-zero).
Version 2.1.8p2
