Post's lattice

In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set, ordered by inclusion. It is named for Emil Post, who published a complete description of the lattice in 1941. The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element set, which has the cardinality of the continuum, and a complicated inner structure. A modern exposition of Post's result can be found in Lau.

Basic concepts

A Boolean function, or logical connective, is an n-ary operation for some, where 2 denotes the two-element set. Particular Boolean functions are the projections
and given an m-ary function f, and n-ary functions g₁,..., g_m, we can construct another n-ary function
called their composition. A set of functions closed under composition, and containing all projections, is called a clone.
Let B be a set of connectives. The functions that can be defined by a formula using propositional variables and connectives from B form a clone , indeed it is the smallest clone that includes B. We call the clone generated by B, and say that B is the basis of . For example, are all Boolean functions, and are the monotone functions.
We use the operations ¬, Np,, ∧, Kpq,, ∨, Apq,, →, Cpq,, ↔, Epq,, +, Jpq, ↛, Lpq,, ?: and the constant unary functions 0 and 1. Moreover, we need the threshold functions
For example, th is the large disjunction of all the variables x_i, and th is the large conjunction. Of particular importance is the majority function
We denote elements of 2ⁿ as vectors:. The set 2ⁿ carries a natural product Boolean algebra structure. That is, ordering, meets, joins, and other operations on n-ary truth assignments are defined pointwise:

Naming of clones

of an arbitrary number of clones is again a clone. It is convenient to denote intersection of clones by simple juxtaposition, i.e., the clone is denoted by C₁C₂...C_k. Some special clones are introduced below:

M = is the set of monotone functions: for every.
D = is the set of self-dual functions:.
A = is the set of affine functions: the functions satisfying
U = is the set of essentially unary functions, i.e., functions that depend on at most one input variable: there exists an i = 1,..., n such that whenever.
Λ = is the set of conjunctive functions:. The clone Λ consists of the conjunctions for all subsets I of , and the constant 0.
V = is the set of disjunctive functions:. Equivalently, V consists of the disjunctions for all subsets I of , and the constant 1.
For any k ≥ 1, T = is the set of functions f such that
For any k ≥ 1, T = is the set of functions f such that
The largest clone of all functions is denoted ⊤ = , the smallest clone is denoted ⊥ = , and = is the clone of constant-preserving functions.
Description of the lattice

clone	one of its bases
⊤	∨, ¬
P₀	∨, +
P₁	∧, →
P	x ? y : z
T₀^k, k ≥ 2	th_k^k+1, ↛
T₀^∞	↛
PT₀^k, k ≥ 2	th_k^k+1, x ∧
PT₀^∞	x ∧
T₁^k, k ≥ 2	th₂^k+1, →
T₁^∞	→
PT₁^k, k ≥ 2	th₂^k+1, x ∨
PT₁^∞	x ∨
M	∧, ∨, 0, 1
MP₀	∧, ∨, 0
MP₁	∧, ∨, 1
MP	∧, ∨
MT₀^k, k ≥ 2	th_k^k+1, 0
MT₀^∞	x ∧, 0
MPT₀^k, k ≥ 2	th_k^k+1 for k ≥ 3, maj, x ∧ for k = 2
MPT₀^∞	x ∧
MT₁^k, k ≥ 2	th₂^k+1, 1
MT₁^∞	x ∨, 1
MPT₁^k, k ≥ 2	th₂^k+1 for k ≥ 3, maj, x ∨ for k = 2
MPT₁^∞	x ∨
Λ	∧, 0, 1
ΛP₀	∧, 0
ΛP₁	∧, 1
ΛP	∧
V	∨, 0, 1
VP₀	∨, 0
VP₁	∨, 1
VP	∨
D	maj, ¬
DP	maj, x + y + z
DM	maj
A	↔, 0
AD	¬, x + y + z
AP₀	+
AP₁	↔
AP	x + y + z
U	¬, 0
UD	¬
UM	0, 1
UP₀	0
UP₁	1
⊥

The eight infinite families have actually also members with k = 1, but these appear separately in the table:,,,,,.
The lattice has a natural symmetry mapping each clone C to its dual clone, where is the de Morgan dual of a Boolean function f. For example,,, and.

Applications

The complete classification of Boolean clones given by Post helps to resolve various questions about classes of Boolean functions. For example:

An inspection of the lattice shows that the maximal clones different from ⊤ are M, D, A, P₀, P₁, and every proper subclone of ⊤ is contained in one of them. As a set B of connectives is functionally complete if and only if it generates ⊤, we obtain the following characterization: B is functionally complete if and only if it is not included in any of the five Post's classes.
The satisfiability problem for Boolean formulas is NP-complete by Cook's theorem. Consider a restricted version of the problem: for a fixed finite set B of connectives, let B-SAT be the algorithmic problem of checking whether a given B-formula is satisfiable. Lewis used the description of Post's lattice to show that B-SAT is NP-complete if the function ↛ can be generated from B, and in all the other cases B-SAT is polynomial-time decidable.
Variants

Clones requiring the constant functions

If one only considers clones that are required to contain the constant functions, the classification is much simpler: there are only 7 such clones: UM, Λ, V, U, A, M, and ⊤. While this can be derived from the full classification, there is a simpler proof, taking less than a page.

Clones allowing nullary functions

Composition alone does not allow to generate a nullary function from the corresponding unary constant function, this is the technical reason why nullary functions are excluded from clones in Post's classification. If we lift the restriction, we get more clones. Namely, each clone C in Post's lattice that contains at least one constant function corresponds to two clones under the less restrictive definition: C, and C together with all nullary functions whose unary versions are in C.

Iterative systems

Post originally did not work with the modern definition of clones, but with the so-called iterative systems, which are sets of operations closed under substitution
as well as permutation and identification of variables. The main difference is that iterative systems do not necessarily contain all projections. Every clone is an iterative system, and there are 20 non-empty iterative systems that are not clones. As another alternative, some authors work with the notion of a closed class, which is an iterative system closed under introduction of dummy variables. There are four closed classes that are not clones: the empty set, the set of constant 0 functions, the set of constant 1 functions, and the set of all constant functions.