Math 2.0

Order, lattices, ordered algebraic structures Lattices: basic notions Problem

Finite lattice representation problem

Builds on LatticeCongruence latticeLattice of subgroupsAlgebraic structureCongruence relation

Problem

Let LL be a finite lattice. Is there always a finite algebra AA whose congruence lattice is isomorphic to LL?

Remarks

Pálfy and Pudlák proved in 1980 that the problem has an equivalent form in group theory. The problem has a positive answer if and only if the following holds. Every finite lattice is isomorphic to an interval [H,G]={K:H≤K≤G}[H, G] = \set{K : H \le K \le G} in the lattice of subgroups of some finite group GG. Here HH is a subgroup of GG, and the interval is ordered by inclusion. The problem is open.

Without the finiteness condition on AA, the answer is yes. By a theorem of Grätzer and Schmidt, every algebraic lattice, and so every finite lattice, is the congruence lattice of some algebra. The difficulty is to find a finite algebra. For a finite distributive lattice the answer is also yes, since it is the congruence lattice of a finite lattice.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Algebraic structure, concept, no evidence levelAlgebraic structureconceptCongruence lattice, concept, no evidence levelCongruence latticeconceptCongruence relation, concept, no evidence levelCongruence relationconceptGroup, concept, no evidence levelGroupconceptLattice, concept, no evidence levelLatticeconceptSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionLattice of subgroups, concept, no evidence levelLattice of subgroupsconceptFinite lattice representation problem, problem, no evidence levelFinite lattice repres…problem
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. Algebraic structure concept
  2. Congruence lattice concept
  3. Congruence relation concept
  4. Group concept
  5. Lattice concept
  6. Subgroup definition
  7. Subgroup criterion statement
  8. Intersection of subgroups is a subgroup statement
  9. Subgroup generated by a subset definition
  10. Lattice of subgroups concept
  11. Finite lattice representation problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_of_subgroups
About this page
TagW83D78WN7T
URIm2://W83D78WN7T
Kindproblem
WritingW0 stub
MSC06B15, 08A30, 20E15
StatusThis copy of the site has no build data. Run m2 status W83D78WN7T.
AboutCongruence lattice, Lattice of subgroups
UsesLattice, Congruence lattice, Lattice of subgroups, Algebraic structure, Congruence relation

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