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### 3 Lattices

#### 3.1 Properties

##### 3.1-1 IsCartesianProset
 ‣ IsCartesianProset( C ) ( property )

Returns: true or false

The property of C being a cartesian proset.

##### 3.1-2 IsMeetSemiLattice
 ‣ IsMeetSemiLattice( C ) ( property )

Returns: true or false

The property of C being a meet semi-lattice.

##### 3.1-3 IsCocartesianProset
 ‣ IsCocartesianProset( C ) ( property )

Returns: true or false

The property of C being a finite cocomplete proset.

##### 3.1-4 IsJoinSemiLattice
 ‣ IsJoinSemiLattice( C ) ( property )

Returns: true or false

The property of C being a join semi-lattice.

##### 3.1-5 IsBicartesianProset
 ‣ IsBicartesianProset( C ) ( property )

Returns: true or false

The property of C being a finite complete and cocomplete proset.

##### 3.1-6 IsLattice
 ‣ IsLattice( C ) ( property )

Returns: true or false

The property of C being a lattice.

##### 3.1-7 IsDistributiveBicartesianProset
 ‣ IsDistributiveBicartesianProset( C ) ( property )

Returns: true or false

The property of C being a distributive (and automatically codistributive) bicartesian proset.

##### 3.1-8 IsDistributiveLattice
 ‣ IsDistributiveLattice( C ) ( property )

Returns: true or false

The property of C being a distributive (and automatically codistributive) lattice.

##### 3.1-9 IsMonoidalLattice
 ‣ IsMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a monoidal lattice.

##### 3.1-10 IsClosedMonoidalLattice
 ‣ IsClosedMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a closed monoidal lattice.

##### 3.1-11 IsCoclosedMonoidalLattice
 ‣ IsCoclosedMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a coclosed monoidal lattice.

##### 3.1-12 IsSymmetricMonoidalLattice
 ‣ IsSymmetricMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a symmetric monoidal lattice.

##### 3.1-13 IsSymmetricClosedMonoidalLattice
 ‣ IsSymmetricClosedMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a symmetric closed monoidal lattice.

##### 3.1-14 IsSymmetricCoclosedMonoidalLattice
 ‣ IsSymmetricCoclosedMonoidalLattice( C ) ( property )

Returns: true or false

The property of C being a symmetric coclosed monoidal lattice.

#### 3.2 Attributes

##### 3.2-1 Factors
 ‣ Factors( A ) ( operation )

Returns: a list

Compute the factors of A. This is a list C with the minimum specification that DirectProduct(L)=A.

##### 3.2-2 FactorsAttr
 ‣ FactorsAttr( arg ) ( attribute )

#### 3.3 Operations

##### 3.3-1 \*
 ‣ \*( A, B ) ( operation )

Returns: an object in a meet semilattice

The meet DirectProduct(A, B) of A and B.

##### 3.3-2 \+
 ‣ \+( A, B ) ( operation )

Returns: an object in a join semilattice

The join Coproduct(A, B) of A and B.

Lattices are thin, finite cartesian (and hence finite complete), finite cocartesian (and hence finite cocomplete), skeletal categories.

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