Post's lattice

In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set {0, 1}, 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 (or larger) 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 (2006).

A Boolean function, or logical connective, is an n-ary operation f: 2ⁿ → 2 for some n ≥ 1, where 2 denotes the two-element set {0, 1}. 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 which can be defined by a formula using propositional variables and connectives from B form a clone [B], indeed it is the smallest clone which includes B. We call [B] the clone generated by B, and say that B is the basis of [B]. For example, [¬, ∧] are all Boolean functions, and [0, 1, ∧, ∨] are the monotone functions.

We use the operations ¬, Np, (negation), ∧, Kpq, (conjunction or meet), ∨, Apq, (disjunction or join), →, Cpq, (implication), ↔, Epq, (biconditional), +, Jpq (exclusive disjunction or Boolean ring addition), ↛, Lpq, (nonimplication), ?: (the ternary conditional operator) and the constant unary functions 0 and 1. Moreover, we need the threshold functions

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