partition-lattice

Partition lattice algebra in exact integer arithmetic: meet, join, and logical entropy. Python bindings for the Rust crate partition-lattice.

A partition of a finite universe splits it into disjoint blocks. Partitions form a lattice under refinement, and this package exposes that lattice directly — the two operations, the order, and the measures — without going through probabilities or floats where integers will do.

from partition_lattice import Partition

a = Partition([0, 0, 1, 1])   # {0,1}{2,3}
b = Partition([0, 1, 0, 1])   # {0,2}{1,3}

a.meet(b).block_count()       # 4 — every element separated
a.join(b).block_count()       # 1 — everything merged
a & b == a.meet(b)            # True; `|` is join
a.logical_entropy()           # 0.5
a.dit_count()                 # 8 — the exact integer numerator

Block labels carry no information, so equal groupings are equal partitions:

Partition([0, 0, 1, 1]) == Partition([7, 7, 3, 3])   # True

Two conventions worth knowing before you start

Order. a.refines(b) means finer, so Partition.discrete(n) is the least element and Partition.indiscrete(n) the greatest. <= is that refinement order.

Meet and join. meet is the common refinement, join the common coarsening. Much of the literature — including Ellerman's papers — uses the order-dual convention, where join is the refinement. Neither is wrong; they are opposite conventions on the same structure. When porting a formula, flip meet/join and the direction of <= together.

The order is partial: a <= b being false does not make a > b true.

Exactness

Distinction counts are integers and stay integers. logical_entropy() is the only place a division happens, and its numerator is available separately as dit_count(), so entropies can be compared exactly rather than through floats.

weighted_entropy(weights) and cross_entropy(p, q) return exact (numerator, denominator) pairs. A weight is a multiplicity, not a probability: the result equals the ordinary logical entropy of the multiset repeating element u exactly w[u] times. Note that a non-uniform weight breaks relabelling invariance.

API

Unlike the Rust from_blocks, Partition.from_blocks rejects a partial or overlapping cover instead of silently reassigning elements.

Install

pip install partition-lattice

Wheels are built against the stable ABI (abi3), so one wheel per platform covers CPython 3.8 and newer.

License

MIT OR Apache-2.0, at your option.