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).
Basic concepts
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, ..., gm, 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.