Interface Lattice
 All Known Implementing Classes:
AbstractDistributiveLattice
,AbstractLattice
,Boolean
,Chain
,Flat
,IntervalAntichains
,Op
,Parts
,Product
,Sum
,Trivial
public interface Lattice
The method isDistributive()
can be used to discover whether distributivity holds.

Field Summary

Method Summary
Modifier and Type Method Description boolean
comp(Element x, Element y)
Return whether the two provided elements are comparable.Map<Element,Set<Element>>
coveringRelation()
Return the covering relation of this lattice.Collection<Element>
elements()
Return all elements of this lattice (optional operation).Collection<Element>
generators()
Return a collection of generators for the lattice.boolean
isDistributive()
Return true if this lattice is distributive.Element
join(Element... element)
Return the join of the provided elements.boolean
leq(Element x, Element y)
Return whether an element is less than or equal to another element in the natural order of this lattice.Element
meet(Element... element)
Return the meet of the provided elements.Element
one()
Return the one of this lattice.Element
pscomp(Element x, Element y)
Return the pseudocomplement of the first element relative to the second element (optional operation).Element
psdiff(Element x, Element y)
Return the Brouwerian pseudodifference of two elements (optional operation).Element
symdiff(Element x, Element y)
Return the symmetric difference of the arguments, that is,psdiff(x,y).join(psdiff(y,x))
.Element
valueOf(String name)
Return an element of this lattice, given its name.Element
zero()
Return the zero of this lattice.

Field Details

UTF8
static final boolean UTF8Use UTF8 symbols for operators in all outputs. This constant is settable using the Boolean system propertyit.unimi.dsi.lama4j.utf8
. 
RING
static final boolean RINGUse ring notation (+
and*
instead of
and&
) in all output. This constant is settable using the Boolean system propertyit.unimi.dsi.lama4j.ring
.


Method Details

isDistributive
boolean isDistributive()Return true if this lattice is distributive. Returns:
 true if this lattice is distributive.

meet
Return the meet of the provided elements. In particular, upon the empty list of arguments returnsone
, and upon a singleton list the only specified element. Parameters:
element
 the elements whose meet has to be computed. Returns:
 the meet of the provided elements.

join
Return the join of the provided elements. In particular, upon the empty list of arguments returnszero
, and upon a singleton list the only specified element. Parameters:
element
 the elements whose join has to be computed. Returns:
 the join of the provided elements.

generators
Collection<Element> generators()Return a collection of generators for the lattice. The set will not includezero
orone
. There is no guarantee of freeness or minimality. Returns:
 a collection of generators.

elements
Collection<Element> elements()Return all elements of this lattice (optional operation).This operation might not implemented, for instance, in infinite lattices.
 Returns:
 the collection of all elements of this lattice.

valueOf
Return an element of this lattice, given its name.Certain lattices make it possible to define names for elements. This method returns the element corresponding to the provided name.
 Parameters:
name
 the name of an element of this lattice. Returns:
 the element of this lattice named
name
.  Throws:
ElementNameException
 if the provided name does not match any element of this lattice.

zero
Element zero()Return the zero of this lattice.Note that there is no guarantee that the returned element is the only element representing zero in this lattice. Other zeroes may arise from computations, but they will always be equal to the element returned by this method.
 Returns:
 the zero of this lattice.

one
Element one()Return the one of this lattice.Note that there is no guarantee that the returned element is the only element representing one in this lattice. Other ones may arise from computations, but they will always be equal to the element returned by this method.
 Returns:
 the one of this lattice.

comp
Return whether the two provided elements are comparable. Parameters:
x
 an element.y
 another element. Returns:
 true if x ≤ y or y ≤ x, false otherwise.

leq
Return whether an element is less than or equal to another element in the natural order of this lattice. Parameters:
x
 an element.y
 another element. Returns:
 true if x ≤ y, false otherwise.

psdiff
Return the Brouwerian pseudodifference of two elements (optional operation).The (Brouwerian) pseudodifference of x and y, usually denoted by x − y, is defined by a Galois connection with the join operation (categorically speaking, an adjunction):
x − y ≤ t iff x ≤ y ∨ t for all t.If the Galois connection exists, this lattice is endowed with the structure of a Brouwerian algebra. In that case, if the lattice is finite
x − y = inf { z  x ≤ y ∨ z },and the lattice is necessarily distributive. Conversely, all finite distributive lattices are Brouwerian algebras, with x − y defined as above.
 Parameters:
x
 an element.y
 another element. Returns:
x
−y
. Throws:
UnsupportedOperationException
 if this lattice is not Browerian.

pscomp
Return the pseudocomplement of the first element relative to the second element (optional operation).The pseudocomplement of x relative to y, denoted by x ⇒ y, is defined by a Galois connection with the meet operation (categorically speaking, an adjunction):
t ≤ y ⇒ x iff x ∧ t ≤ y for all t.If the Galois connection exists, this lattice is endowed with the structure of a Heyting algebra. In that case, if the lattice is finite
x ⇒ y = sup { z  x ∧ z ≤ y },and the lattice is necessarily distributive. Conversely, all finite distributive lattices are Heyting algebras.
 Parameters:
x
 an element.y
 another element. Returns:
x
⇒y
.

symdiff
Return the symmetric difference of the arguments, that is,psdiff(x,y).join(psdiff(y,x))
. Parameters:
x
 an element.y
 another element. Returns:
x
Δy
. See Also:
Element.join(Element)
,psdiff(Element, Element)

coveringRelation
Return the covering relation of this lattice.The covering relation of a lattice relates elements x, y such that there is no element strictly between x and y. In can be interpreted as a graph and drawn, resulting in the Hasse diagram of the lattice.
 Returns:
 a map from elements to set of elements representing the covering relation.
