it.unimi.dsi.lama4j

Class Product

java.lang.Object

it.unimi.dsi.lama4j.AbstractLattice

it.unimi.dsi.lama4j.Product

All Implemented Interfaces:

Lattice

public class Product
extends AbstractLattice

The (categorical) product of lattices. As for every algebraic theory, elements are tuples, and operations are implemented componentwise.

Note that the product of zero lattices is the trivial lattice with one element.

Field Summary

Fields inherited from interface it.unimi.dsi.lama4j.Lattice

RING, UTF8

Method Summary

Modifier and Type	Method and Description
boolean	comp(Element x, Element y) Check whether two elements are comparable using Lattice.meet(Element[]).
Lattice	component(int k) Return a component lattice, given its index.
Collection<Element>	elements() Generate iteratively all elements of this lattice.
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) Check whether an element is smaller than or equal to another element using Lattice.meet(Element[]).
Element	meet(Element... element) Return the meet of the provided elements.
static Lattice	of(Lattice... lattice) Return the product of the provided lattices.
Element	one() Return the one of this lattice.
Element	proj(Element e, int i) Project the provided element on a component specified by index.
String	toString()
Element	tuple(Element... element) Return an element of this product given a tuple of elements from its factors.
Element	valueOf(String name) Return an element of this lattice, given its name.
Element	zero() Return the zero of this lattice.

Methods inherited from class it.unimi.dsi.lama4j.AbstractLattice

coveringRelation, ensureElementsInLattice, ensureElementsInLattice, ensureElementsInLattice, pscomp, psdiff, symdiff, valueOfZeroOrOne

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Method Detail

of

public static Lattice of(Lattice... lattice)

Return the product of the provided lattices.

Parameters:

lattice - the lattices whose product has to be computed.

Returns:

the product of the specified lattices.

isDistributive

public boolean isDistributive()

Description copied from interface: Lattice

Return true if this lattice is distributive.

Specified by:

isDistributive in interface Lattice

Overrides:

isDistributive in class AbstractLattice

Returns:

true if this lattice is distributive.

elements

public Collection<Element> elements()

Description copied from class: AbstractLattice

Generate iteratively all elements of this lattice.

This methods uses Lattice.generators() to obtain an initial set of elements, and then computes joins and meets of all available elements until no new elements are generated. It is expected that concrete subclasses will override this method with an ad hoc, more efficient implementation.

It is strongly suggested that concrete subclasses that do not override this method cache its result internally.

Specified by:

elements in interface Lattice

Overrides:

elements in class AbstractLattice

Returns:

the set of elements of this lattice.

generators

public Collection<Element> generators()

Description copied from interface: Lattice

Return a collection of generators for the lattice. The set will not include zero or one. There is no guarantee of freeness or minimality.

Returns:

a collection of generators.

proj

public Element proj(Element e,
                    int i)

Project the provided element on a component specified by index.

Parameters:

e - an element of this product.

i - a component, specified by its index starting at 0.

Returns:

the projection of e on the i-th component.

valueOf

public Element valueOf(String name)

Description copied from interface: Lattice

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.

tuple

public Element tuple(Element... element)

Return an element of this product given a tuple of elements from its factors.

Parameters:

element - a list of elements from each factor of this product, in the same order.

Returns:

the element of the product defined by the tuple element.

one

public Element one()

Description copied from interface: Lattice

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.

zero

public Element zero()

Description copied from interface: Lattice

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.

join

public Element join(Element... element)

Description copied from interface: Lattice

Return the join of the provided elements. In particular, upon the empty list of arguments returns zero, 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.

meet

public Element meet(Element... element)

Description copied from interface: Lattice

Return the meet of the provided elements. In particular, upon the empty list of arguments returns one, 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.

component

public Lattice component(int k)

Return a component lattice, given its index.

The component lattices of a product are numbered from 0. This method provides access by index to such components.

Parameters:

k - a component index.

Returns:

the component lattice of index k.

comp

public boolean comp(Element x,
                    Element y)

Description copied from class: AbstractLattice

Check whether two elements are comparable using Lattice.meet(Element[]).

Specified by:

comp in interface Lattice

Overrides:

comp in class AbstractLattice

Parameters:

x - an element.

y - another element.

Returns:

true iff x.meet(y) equals x or y.

leq

public boolean leq(Element x,
                   Element y)

Description copied from class: AbstractLattice

Check whether an element is smaller than or equal to another element using Lattice.meet(Element[]).

Specified by:

leq in interface Lattice

Overrides:

leq in class AbstractLattice

Parameters:

x - an element.

y - another element.

Returns:

true iff x.meet(y) equals x.

toString

public String toString()

Overrides:

toString in class Object