Math 2.0

General algebraic systems Clones and operations Object

Post's lattice

Builds on CloneBoolean function

Description

Post's lattice is the lattice of clones on the two-element set {0,1}\set{0, 1}, that is, the set of all clones of Boolean functions ordered by inclusion. Emil Post gave a complete description of it, published in 1941. Its main features are these.

  1. It is countably infinite.
  2. Every clone on {0,1}\set{0, 1} is generated by finitely many Boolean functions.
  3. It has exactly five maximal clones, that is, coatoms below the clone of all Boolean functions. They consist of the operations that preserve 00, that preserve 11, that are monotone, that are self-dual, and that are affine.
  4. Its least element is the clone of projections.

Check of one maximal clone: the monotone operations form a clone, since projections are monotone and a composition of monotone operations is monotone. It is a proper clone, since negation ¬\neg is not monotone.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Boolean function, concept, no evidence levelBoolean functionconceptClone, concept, no evidence levelCloneconceptPost's lattice, object, no evidence levelPost's latticeobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boolean function concept
  2. Clone concept
  3. Post's lattice object

Further reading

  1. Web https://en.wikipedia.org/wiki/Post%27s_lattice
About this page
TagMA20X7JCET
URIm2://MA20X7JCET
Kindobject
WritingW0 stub
MSC08A40, 06E30, 03B50
Instance ofLattice of clones
UsesClone, Boolean function
Same as
Changed2 November 2026

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