Op (LaMa4J 1.1.1)

java.lang.Object
- it.unimi.dsi.lama4j.Op

All Implemented Interfaces:

Lattice
```
public class Op
extends Object
implements Lattice
```
The opposite (a.k.a. dual) of a given lattice.
Instances of this class wrap a given lattice, dualising all its operations.
Since the lattice axioms (including distributivity) are self-dual, by reversing the order relation (or, equivalently, exchanging meets and joins) we get a new lattice. Note that the definitions of difference and relative pseudocomplement are dual, so a lattice is a Browerian algebra iff its opposite is a Heyting algebra.

Field Summary
- Fields inherited from interface it.unimi.dsi.lama4j.Lattice
  RING, UTF8

Constructor Summary

Constructors
Constructor and Description

Op(Lattice lattice)

Constructors
Constructor and Description
`Op(Lattice lattice)`

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`boolean`	`comp(Element x, Element y)` Return whether the two provided elements are comparable.
`Map<Element,Set<Element>>`	`coveringRelation()` Compute the covering relation by transposing the covering relation of the dualised 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 pseudo-difference 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 a dualised element by name.
`Element`	`zero()` Return the zero of this lattice.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - Op
```
public Op(Lattice lattice)
```
- Method Detail
  - isDistributive
```
public boolean isDistributive()
```
    Description copied from interface: Lattice
    
    Return true if this lattice is distributive.
    
    Specified by:
    
    isDistributive in interface Lattice
    
    Returns:
    
    true if this lattice is distributive.
  - 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.
    
    Specified by:
    
    meet in interface Lattice
    
    Parameters:
    
    element - the elements whose meet has to be computed.
    
    Returns:
    
    the meet of the provided elements.
  - 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.
    
    Specified by:
    
    join in interface Lattice
    
    Parameters:
    
    element - the elements whose join has to be computed.
    
    Returns:
    
    the join of the provided elements.
  - 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.
    
    Specified by:
    
    generators in interface Lattice
    
    Returns:
    
    a collection of generators.
  - elements
```
public Collection<Element> elements()
```
    Description copied from interface: Lattice
    
    Return all elements of this lattice (optional operation).
    This operation might not implemented, for instance, in infinite lattices.
    
    Specified by:
    
    elements in interface Lattice
    
    Returns:
    
    the collection of all elements of this lattice.
  - valueOf
```
public Element valueOf(String name)
```
    Return a dualised element by name.
    Dual elements have the same name of the corresponding elements of the dualised lattice, except for zero and one, which have the standard names.
    
    Specified by:
    
    valueOf in interface Lattice
    
    Parameters:
    
    name - the name of an element of this lattice.
    
    Returns:
    
    the element of this lattice named name.
  - 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.
    
    Specified by:
    
    zero in interface Lattice
    
    Returns:
    
    the zero of this lattice.
  - 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.
    
    Specified by:
    
    one in interface Lattice
    
    Returns:
    
    the one of this lattice.
  - comp
```
public boolean comp(Element x,
                    Element y)
```
    Description copied from interface: Lattice
    
    Return whether the two provided elements are comparable.
    
    Specified by:
    
    comp in interface Lattice
    
    Parameters:
    
    x - an element.
    
    y - another element.
    
    Returns:
    
    true if x ≤ y or y ≤ x, false otherwise.
  - leq
```
public boolean leq(Element x,
                   Element y)
```
    Description copied from interface: Lattice
    
    Return whether an element is less than or equal to another element in the natural order of this lattice.
    
    Specified by:
    
    leq in interface Lattice
    
    Parameters:
    
    x - an element.
    
    y - another element.
    
    Returns:
    
    true if x ≤ y, false otherwise.
  - psdiff
```
public Element psdiff(Element x,
                      Element y)
```
    Description copied from interface: Lattice
    
    Return the Brouwerian pseudo-difference of two elements (optional operation).
    The (Brouwerian) pseudo-difference 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.
    
    Specified by:
    
    psdiff in interface Lattice
    
    Parameters:
    
    x - an element.
    
    y - another element.
    
    Returns:
    
    x − y.
  - pscomp
```
public Element pscomp(Element x,
                      Element y)
```
    Description copied from interface: Lattice
    
    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.
    
    Specified by:
    
    pscomp in interface Lattice
    
    Parameters:
    
    x - an element.
    
    y - another element.
    
    Returns:
    
    x ⇒ y.
  - symdiff
```
public Element symdiff(Element x,
                       Element y)
```
    Description copied from interface: Lattice
    
    Return the symmetric difference of the arguments, that is, psdiff(x,y).join(psdiff(y,x)).
    
    Specified by:
    
    symdiff in interface Lattice
    
    Parameters:
    
    x - an element.
    
    y - another element.
    
    Returns:
    
    x Δ y.
    
    See Also:
    
    Element.join(Element), Lattice.psdiff(Element, Element)
  - coveringRelation
```
public Map<Element,Set<Element>> coveringRelation()
```
    Compute the covering relation by transposing the covering relation of the dualised lattice.
    
    Specified by:
    
    coveringRelation in interface Lattice
    
    Returns:
    
    a map from elements to set of elements representing the covering relation.

Class Op

Field Summary

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

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

Op

Method Detail

isDistributive

meet

join

generators

elements

valueOf

zero

one

comp

leq

psdiff

pscomp

symdiff

coveringRelation