Semilattices

Definition

A semilattice is a commutative semigroup

A (meet) semilattice is a partially ordered set

Basic Results

• Every semilattice is a meetsemilattice (Interactive PCP), (complete PCProof)
• Every meetsemilattice is a semilattice (Interactive PCP), (complete PCProof)
• By duality, every semilattice is a joinsemilattice and vice versa.

Examples

• Any lattice
• Some semilattices that aren't lattices:
  • this minimal 3-element example:

                        * | 0 1 2
                        ---------
                        0 | 0 0 0
                        1 | 0 1 0
                        2 | 0 0 2
                      

Representation

Any semilattice is isomorphic to a semilattice of subsets of some set, under the operation of union (or intersection). In fact a meetsemilattice is isomorphic to it's lattice of principal ideals, under intersection.

Decision problems

Equational Theory: Decidable

Quasi-equational Theory: Decidable

First order theory: Undecidable

Spectra and growth

Finite spectrum:

Free spectrum:

Subvarieties

only the variety of one-element groupoids

Subclasses

reducts of lattices

Expansions

lattices

semilattice ordered algebraic structures

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PCP: Point and Click Proofs, Peter Jipsen, 2001

Adapted from J.Pedersen's Catalogue of Algebraic Systems