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#### Abbreviation: Slat

### Definition

A *semilattice* is a structure *S* = (*S*,·), where · is an infix binary operation, called the
*semilattice operation*, such that

· is associative: (*x**y*)*z* = *x*(*y**z*),

· is commutative: *x**y* = *y**x* and

· is idempotent: *x**x* = *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 *j**o**i**n*, 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 *m**e**e**t*, such that

≤ is a partial order, where *x* ≤ *y* ⇔ *x*∧*y* = *x*and

*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*(*x**y*) = *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

[Size 1]?: 1

[Size 2]?: 1

[Size 3]?: 2

[Size 4]?: 5

[Size 5]?: 15

[Size 6]?: 53

[Size 7]?: 222

[Size 8]?: 1078

[Size 9]?: 5994

[Size 10]?: 37622

[Size 11]?: 262776

[Size 12]?: 2018305

[Size 13]?: 16873364

[Size 14]?: 152233518

[Size 15]?: 1471613387

[Size 16]?: 15150569446

[Size 17]?: 165269824761

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

[One-element algebras]?

Semilattices with zero

Semilattices with identity

### Superclasses

Commutative semigroups

[Partial semilattices]?