For being able to conceive an objectual
definition of a term, we shall start with an *objectual *analysis
(synonym - *systemic*) of the definitions assigned to this word
by dictionaries. The objectual approach implies the extraction of the
common component from a set of abstract objects, this component
becoming the class model for the generic abstract object of this set
(notion). According to the Dictionary of General Mathematics^{9}1
we find out that:

The

*complement*term comes from the latin word*complementum*(completeness, replenishment);

These
are few excerpts from The Encyclopedic Dictionary^{9}2:

*Complementary*: what is added to something in order to replenish it;The

*complement*of a number with*n*digits written within a numerical system with the base*q*is the difference between*q*^{n}and that number;Two angles are

*complementary*if their sum is ;The

*complementary*of a set*A*against another set*B*is the set of the elements which do not belong to*A*but belong to*B*;Two colors which belong to the visible spectrum are

*complementary*if when they are superposed, white color occurs;

At last, The
Dictionary of Logics^{9}3
issues the following definition:

*Complementary:*“operation which, by starting from a set*X*, makes possible the creation of another set (*nonX*or**C***X*) named*complementary set*and defined as follows: . It is assumed that*X*is taken from an universe*U*, so that (the operator [+] stands for*exclusion*in this context). By means of**c**., we divide the universe in two classes (dichotomy).**C.**has the property of*involution**()*, and the intersection between and*X*is void”.

Comment
X.5.1.1: By using specific terms for this paper, the operator [+]
symbolizes the adjacent-disjoint union of the two sets, so that the
existence domain of the universe *U *to be equal with the sum of
the domains belonging to the sets *X* and
.
It is clear that the disjunction relation implies the exclusion as
well.

Taking into account
the seven definitions which were previously presented, now, we shall
extract the common components with which we shall design the generic
model of the abstract object *complementarity*. First, we must
notice that this complementarity involves many *relations*
between __three__ abstract objects (an object which is considered
*as whole *and the two sections in which it is divided), which
are relations underlined at the points 3, 4, 5, 6 and mostly 7. These
relations determine the bipartition (splitting in two parts, the
dichotomy) of a whole object named * base*, namely, other
two objects which claim the internal domain of this base.

Comment
X.5.1.2: The *base* term for the existence range of the
complementarity relation must not be mistaken with the *base*
term of the *numbering system* which may be found at the
definition from the point 3. In this case (from the point 3), the
complementarity base is represented by the term *q*^{n}
, whereas the base of the numbering system is *q.* In case of
the definition from the point 7, the base of complementarity
corresponds with the universe *U.*

Each of the two
objects which were generated by means of the base splitting is the
complement of the other __against the common base__, the union of
their individual domains being obviously equal by definition with the
internal domain of the base.

Therefore, the
*complementarity* is a *complex relation* (decomposable)
deployed between two abstract objects whose internal domains
represent a bipartition of another object (base), and the following
elementary relations with simultaneous existence belong to this
relation:

Bipartition relation - union (sum) of the internal domains of the two complementary objects is equal with the domain of the base object;

Disjunction relation - intersection (conjunction) of the two domains is void;

The adjacency relation - there is a common boundary between the two domains.

The disjunction relation between the internal domains of the two complementary objects involves the exclusion of the attribution of a singular property value to both objects. On the other hand, one may observe that an object with a specific qualitative property (number, angle, wave length, set of objects belonging to the same class etc.) distributed on its unitary support domain (base) is divided in two abstract objects which own the same property, objects which are assigned with different qualitative attributes (for instance, positive and negative), although the single difference between them is only the support domain on which this unique property is distributed (complementary domains generated as a result of the base bipartition).

Comment
X.5.1.3: There is a special case of complementarity which apparently
does not comply with the definitions stated by the objectual
philosophy, that is the complementarity with a null base. As it was
mentioned in chapters 1…9 of the present paper, an abstract
object with a null existential attribute means that it does not
exist. In case of a null base, it seems that no complementariness is
able to exist either. However, if there are two qualitative
properties with non-zero existential attribute, belonging to two
different objects which are able to form a complex object but which
properties are no longer existent in case of the complex object (as
if they were mutually cancelled), those properties are also
considered as complementary. For instance, in case of EP with
opposite charges (a proton and an electron), the complex object from
the two EPs (e.g.: the hydrogen atom) does not have charge attributes
__in the outside__. In this case, the complementariness base is
represented by the complex object which has null properties in terms
of the electric charge.

91
*Dicționar de Matematici Generale* - Editura Enciclopedică -
1974

92*
Dicționar Enciclopedic* - Editura Enciclopedică - 1993…1999

93*
Dicționar de Logică* - Editura Științifică și Enciclopedică -
1985

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