Semilattices

Abbreviation: Slat

Definition

A semilattice is a structure S = (S,·), where · is an infix binary operation, called the semilattice operation, such that
· is associative:   (xy)z = x(yz),
· is commutative:   xy = yx and
· is idempotent:   xx = x.

Remark: This definition shows that semilattices form a variety.

Definition

A join-semilattice is a structure S = (S,∨), where ∨ is an infix binary operation, called the join, such that
 ≤  is a partial order, where x ≤ y  ⇔   x∨y = y and
x∨y is the least upper bound of {x,y}.

Definition

A meet-semilattice is a structure S = (S,∧), where ∧ is an infix binary operation, called the meet, such that
 ≤  is a partial order, where x ≤ y  ⇔   x∧y = xand
x∧y is the greatest lower bound of {x,y}.

Morphisms

Let S and T be semilattices. A morphism from S to T is a function h : S→T that is a homomorphism: h(xy) = h(x)h(y).

Some results

Examples

(P_<i>ω(X)−{Ø},∪), the set of finite nonempty subsets of a set X, with union, is the free join-semilattice with singleton subsets of X as generators.

Properties

Finite members

These results follow from the paper below and the observation that semilattices with n elements are in 1-1 correspondence to lattices with n + 1 elements.

[Jobst Heitzig, Jürgen Reinhold, Counting finite lattices, Algebra Universalis 48 (2002) 43--53 MRreview]

Search for finite semilattices

Subclasses

Superclasses