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With Sets
and Collectors
the semantics of operator 
is that of set difference which means, that you can
only subtract what has been put into the container before. With Quantifiers
that count
or quantify their key
values in some way, the semantics of operator

may be different.
The question is how subtraction should be defined here?
//Pseudocode: icl::map<int,some_number> q = {(1>1)}; q = (2>1);
If type some_number
is unsigned
a set difference
kind of subtraction make sense
icl::map<int,some_number> q = {(1>1)}; q = (2>1); // key 2 is not in the map so q == {(1>1)}; // q is unchanged by 'aggregate on collision'
If some_number
is a signed
numerical type the result can also
be this
icl::map<int,some_number> q = {(1>1)}; q = (2>1); // subtracting works like q == {(1>1), (2> 1)}; // adding the inverse element
As commented in the example, subtraction of a key value pair (k,v)
can
obviously be defined as adding the inverse
element for that key (k,v)
, if the key is not yet stored in the map.
Another concept, that we can think of, is that in a Quantifier
every key_value
is initially
quantified 0
times, where 0
stands for the neutral element of the numeric
CodomainT
type. Such a Quantifier
would be totally defined on
all values of it's DomainT
type and can be conceived as an InfiniteVector
.
To create an infinite vector that is totally defined on it's domain we can
set the map's Trait
parameter
to the value total_absorber
.
The total_absorber
trait fits specifically well with a Quantifier
if it's CodomainT
has an
inverse element, like all signed numerical type have. As we can see later
in this section this kind of a total Quantifier
has the basic properties that elements of a vector
space do provide.
Another difference between Collectors
and Quantifiers
is the semantics
of operator &
,
that has the meaning of set intersection for Collectors
.
For the aggregate on overlap principle the operation
&
has to be passed to combine
associated values on overlap of intervals or collision of keys. This can
not be done for Quantifiers
,
since numeric types do not implement intersection.
For CodomainT
types that
are not models of Sets
operator &
is defined as aggregation on the intersection of the domains.
Instead of the codomain_intersect
functor codomain_combine
is used as aggregation operation:
//Pseudocode example for partial Quantifiers p, q: interval_map<int,int> p, q; p = {[1 3)>1 }; q = { ([2 4)>1}; p & q =={ [2 3)>2 };
So an addition or aggregation of associated values is done like for operator +
but value pairs that have no common keys are not added to the result.
For a Quantifier
that is
a model of an InfiniteVector
and which is therefore defined for every key value of the DomainT
type, this definition of operator
&
degenerates to the same sematics
that operaotor +
implements:
//Pseudocode example for total Quantifiers p, q: interval_map<int,int> p, q; p = {[min 1)[1 3)[3 max]}; >0 >1 >0 q = {[min 2)[2 4)[4 max]}; >0 >1 >0 p&q =={[min 1)[1 2)[2 3)[3 4)[4 max]}; >0 >1 >2 >1 >0
The semantics of icl Maps of Numbers is different for unsigned or signed
numbers. So the sets of laws that are valid for Quantifiers will be different
depending on the instantiation of an unsigned or a signed number type as
CodomainT
parameter.
Again, we are presenting the investigated sets of laws, this time for Quantifier
types Q
which are icl::map
<D,N,T>
, interval_map
<D,N,T>
and split_interval_map
<D,N,T>
where CodomainT
type N
is a Number
and Trait
type T
is one of the icl's
map traits.
Associativity<Q,+,== >: Q a,b,c; a+(b+c) == (a+b)+c Neutrality<Q,+,== > : Q a; a+Q() == a Commutativity<Q,+,== >: Q a,b; a+b == b+a Associativity<Q,&,== >: Q a,b,c; a&(b&c) ==(a&b)&c Commutativity<Q,&,== >: Q a,b; a&b == b&a RightNeutrality<Q,,== >: Q a; aQ() == a Inversion<Q,,=v= > : Q a; a  a =v= Q()
For an unsigned Quantifier
,
an icl Map of unsigned numbers
,
the same basic laws apply that are valid for Collectors
:
+ &  Associativity == == Neutrality == == Commutativity == == Inversion absorbs_identities == enriches_identities =d=
The subset of laws, that relates to operator
+
and the neutral element Q()
is
that of a commutative monoid. This is the same concept, that applies for
the CodomainT
type. This
gives rise to the assumption that an icl Map
over a CommutativeModoid
is again a CommutativeModoid
.
Other laws that were valid for Collectors
are not valid for an unsigned Quantifier
.
For Quantifiers
of signed
numbers, or signed Quantifiers
,
the pattern of valid laws is somewhat different:
+ &  Associativity =v= =v= Neutrality == == Commutativity == == Inversion absorbs_identities == enriches_identities =d=
The differences are tagged as =v=
indicating,
that the associativity law is not uniquely valid for a single equality relation
==
as this was the case for
Collector
and unsigned Quntifier
maps.
The differences are these:
+ Associativity icl::map == interval_map == split_interval_map =e=
For operator +
the associativity on split_interval_maps
is only valid with element equality =e=
, which
is not a big constrained, because only element equality is required.
For operator &
the associativity is broken for all maps that are partial absorbers. For
total absorbers associativity is valid for element equality. All maps having
the identity enricher Trait are associative wrt. lexicographical
equality ==
.
Associativity & absorbs_identities && !is_total false absorbs_identities && is_total =e= enriches_identities ==
Note, that all laws that establish a commutative monoid for operator +
and identity element Q()
are valid for signed Quantifiers
.
In addition symmetric difference that does not hold for unsigned
Qunatifiers
is valid for signed Qunatifiers
.
SymmetricDifference<Q,== > : Q a,b,c; (a + b)  (a & b) == (a  b) + (b  a)
For a signed TotalQuantifier
Qt
symmetrical difference
degenerates to a trivial form since operator
&
and operator
+
become identical
SymmetricDifference<Qt,== > : Qt a,b,c; (a + b)  (a + b) == (a  b) + (b  a) == Qt()
By now signed Quantifiers
Q
are commutative monoids
with respect to the operator +
and the neutral element Q()
. If the Quantifier's CodomainT
type has an inverse element like e.g. signed numbers
do, the CodomainT
type is
a commutative or abelian group. In this case a signed Quantifier
that is also total
has an inverse and
the following law holds:
InverseElement<Qt,== > : Qt a; (0  a) + a == 0
Which means that each TotalQuantifier
over an abelian group is an abelian group itself.
This also implies that a Quantifier
of Quantifiers
is again a
Quantifiers
and a TotalQuantifier
of TotalQuantifiers
is also a TotalQuantifier
.
TotalQuantifiers
resemble
the notion of a vector space partially. The concept could be completed to
a vector space, if a scalar multiplication was added.