Map of lattices

The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices.

Diagram showing implications of common lattice propertiesPosetSemilatticeLattice (order)Relatively complementedBoolean algebra (structure)Atomic latticeSemimodular latticeModular latticeDistributive latticeTotal orderMetric latticeProjective latticeGeometric latticeResiduated latticeHeyting algebraBounded latticeComplemented latticeOrthocomplemented latticeOrthomodular latticeComplete latticeAlgebraic lattice
Diagram showing implications of common lattice properties