Mathematical introduction
In the following, a mathematical structure is introduced that helps to
describe the sexual reproduction between two individuals. For this purpose
the set L is introduced, which is the set of all possible
creatures with sexual reproduction (see Fig. 1). “All possible” refers to
existent, extinct, future, realized, and never realized but principally
possible. The xth element of this set will be called lx. The union of
all species with sexual reproduction is a subset of L (see Eq. 1).
It is just a subset, because not all creatures need to belong to a species.
If for some reason it turns out that all creatures belong to a species the
term “subset” includes equality anyhow. Species i will be denoted by
Si, e.g., Shazel, Sdog,
Slion and so on. In the following natural numbers are used, starting
with 1, instead of words; hence, i∈N. Here, S1
represents the set of all hazels, S2 that of all dogs, and so on.
Within this paper, only gonochoric reproduction with
two sexes will be considered. In gonochoric reproduction, offspring is
created via sex between a fertile male individual and a fertile female individual.
Therefore, the subsets Lm, Smi and Lf,
Sfi are introduced, meaning fertile male and fertile female
individuals (for multi-sexual reproduction, such a set would have to be
introduced for every sex). The set Lm contains all possible
fertile male creatures and Smi all possible fertile male
individuals belonging to a certain species i. The same respective
designation is used for the females. Additionally, infertile individuals are
considered via Li and Sii. It is important to note that
only individuals that do not have the potential to become fertile (i.e., they
are necessarily infertile during their whole life) belong to the sets
Li or Sii (e.g., individuals which lack gonads). For the
sake of simplicity, it is additionally assumed that every individual has a
single sex during its entire life (i.e., no switching between fertile male and
fertile female or vice versa is possible). This means that a single individual
can be an element of Smi, Sfi, or Sii.
The xth element of the corresponding sets will be called sxi,
smxi, sfxi, and sixi, for example. Therefore, the
following relations hold:
L=Lm∪Lf∪Li⊇S1∪S2∪…∪SnLm⊇Sm1∪Sm2∪…∪SmnLf⊇Sf1∪Sf2∪…∪SfnLi⊇Si1∪Sf2∪…∪SinSi=Smi∪Sfi∪SiiL={l1,l2,l3,…,lh}Si={s1i,s2i,s3i,…,sgii}Smi={smi1,smi2,smi3,…,smiki}Sfi={sfi1,sfi2,sfi3,…,sfili}Sii={si1i,si2i,si3i,…,simii}
A set of individuals belonging to species Si can be
divided into subsets Bji. All individuals belonging to a certain
subset Bji belong to the same breed.
![]()
In Eq. (), n∈N is the number of all possible
species, h∈N is the number of all possible life-forms, gi∈N is the number of all possible individuals belonging to
species i, ki∈N is the number of all possible fertile male
individuals belonging to species i, li∈N is the number of
all possible fertile female individuals belonging to species i, and mi∈N is the number of all possible sterile individuals belonging
to species i. The number of elements of set Si is equal to all
elements that belong to the union of Smi, Sfi, and
Sii. Therefore, gi=ki+li+mi.
From the introduction we know that a breed is a subset of a certain species.
For the species Si, i∈N, breeds are denoted by
Bji, j∈N (see Fig. 2 and Eq. ). The
superscript i denotes the species (e.g., dog) and the subscript j denotes
the specific breed (e.g., whippet). Here, a distinction is also made between fertile
male, fertile female, and sterile individuals. Therefore, we can write
Si⊇B1i∪B2i∪…∪BqiiBji=Bmji∪Bfji∪Biji
In Eq. (), qi∈N is the number of all breeds
belonging to species i. Here, it is clear that the union of all breed sets
is just a subset of the corresponding species set, because it is evident that
e.g., not all dogs belong to a breed. The breed set Bji consists
of the union of all fertile males, fertile females and infertile individuals
that fulfill the corresponding conditions. In principle, it is possible that
a certain individual from species i is an element of several breed sets
(compare with Fig. 2).
In order to consider the process of gonochoric reproduction and to formulate
the term “breed” via a mathematical equation, two operators, i.e.,
“∘” and “•”, are introduced. Commonly, in mathematics an
operator, e.g., “+”, combines two elements and gives a third element,
e.g., a+b=c . To describe the process of gonochoric reproduction of
two individuals such a structure is insufficient, as gonochoric parents
can potentially have a large number of offspring; furthermore each offspring may be
different. Therefore, an operator is needed that combines two elements of the
set L, e.g., lx∘ly, and the result is the set of all
possible offspring between these two elements, such as {lx,y,1,lx,y,2,,lx,y,3}. Additionally, we define an operator which also gives
the corresponding set of probabilities for each offspring to be realized:
lx∘ly=Lx,ylx•ly=Px,y
The set Lx,y consists of all possible offspring between an
individual lx and an individual ly: Lx,y={lx,y,1,lx,y,2,,…}. The set Px,y consists of pairs
(lx,y,i,px,y,i), where px,y,i is the corresponding probability
value, i.e., how probable it is that offspring lx,y,i will be realized.
The probability is element of
the rational numbers, i.e., px,y,i∈Q. This means that
offspring lx,y,1 is realized with a probability of px,y,1 and so
on. Therefore, the set Px,y takes the following form:
Px,y={(lx,y,1,px,y,1),(lx,y,2,px,y,2),…}. The elements as well as the corresponding probability values have to be
determined according to laws of heredity. How to determine the elements of
Lx,y and Px,y in practice is shown by a simple
example in Sect. 4.
(L,∘,•) is a mathematical structure. The operator
∘ maps two elements of set L onto the set of possible
offspring (see Eq. ). Let us name this set O,
for offspring. It contains the sets for all possible offspring combinations
of the elements of L: O={…,Lx,y,Lx,y+1,…}={{l1},…,{l1,l2},…,{l1,l2,l3},…}. Hence, it
contains sets with single elements, with two elements, with three elements, and so on in all
possible combinations. The operator • maps two elements of set
L onto the set P =
{…,Px,y,Px,y+1,…} = {{(l1,p1)},…,{(l1,p1),(l2,p2)},…}, which contains doublets of the
form L×Q (see Eq. ).
∘:L×L→O•:L×L→P
Both operators are commutative. This means that lx∘ly=ly∘lx and lx•ly=ly•lx. A notation like lx∘(ly∘lz) makes no sense as the term in brackets is a set and both
operators can only be applied to single elements of L. Therefore,
the introduced structure is not associative, which is typical for structures
that describe inheritance, e.g., genetic algebras
e.g.,. The
introduced structure only allows for the combination of two elements of the
set L. There is no neutral or inverse element. A neutral element
fulfills the equation n∘l=l. However, even for asexual cloning the
equation looks like l∘l={l}. Therefore, from a mathematical point
of view the introduced structure is boring. However, it may be used to give a
set-theoretic definition of the term “breed”.
For most combinations of lx and ly the resulting sets,
Lx,y and Px,y, will be empty sets. For example,
in the case where lx is an element of Lm and ly is an
element of Lf, but lx and ly do not belong to the same
species. (In common definitions of the term
“species”, such as Mayr, 2003, individuals of different species may produce
(infertile) offspring. For example horses and donkeys, or lions and tigers.
For simplicity, in this paper the term “species” is used in such a way that
it obeys the first row of Eq. (). This stands in
contradiction to common definitions based on words. However, this will not
have any influence on the mathematical definition of the term “breed”. In
principle one could name the sets that obey Eq. () with a
new term, e.g., “hyperspecies”. However, as long as the terms “species”
and “subspecies” lack mathematical definitions it probably makes no sense
to introduce new terms.) In the following, we consider elements lx that
are elements of some set Si and will write smxi,
sfxi, or sixi instead. In Eq. () all
combinations that result in the empty set are listed:
smxi∘sfyj={},ifi≠jsmxi∘smyj={},∀i,jsfxi∘sfyj={},∀i,jsixi∘siyj={},∀i,jsmxi∘siyj={},∀i,jsfxi∘siyj={},∀i,j
Equation () expresses the following: the first row states that fertile males
and fertile females belonging to different species can not produce any
offspring. The following rows state that two fertile male individuals, two
fertile female individuals, two infertile individuals, etc. can not have any offspring, even
if they are from the same species.
Only in cases where the operator “∘” is applied to individuals between
Smi and Sfi, does a set unequal to the empty set result:
smxi∘sfyi=Smx,yi∪Sfx,yi∪Six,yi=Sx,yiSx,yi=sx,y,1i,sx,y,2i,sx,y,3i,…,sx,y,ai,…,sx,y,wx,y,iiSmx,yi=smx,y,1i,smx,y,2i,…,smx,y,ai,…,smx,y,rx,y,iiSfx,yi=sfx,y,1i,sfx,y,2i,…,sfx,y,ai,…,sfx,y,tx,y,iiSix,yi=six,y,1i,six,y,2i,…,six,y,ai,…,six,y,ox,y,ii
In Eq. () the genotype of the fertile male individual
smxi and the fertile female individual sfyi
determine all possible genotypes of the common offspring, e.g.,
sx,y,1i. The number of all possible offspring between the two
individuals smxi and smyi is wx,y,i∈N. The sum of Smix,y plus
Sfix,y plus
Siix,y is equal to |Sx,yi|.
Therefore: rx,y,i+tx,y,i+ox,y,i=wx,y,i. For completeness,
it is noted that not all elements of Sx,yi are necessarily
elements of Si. Otherwise, it would not be possible for new
species to arise over time. All elements of Sx,yi will be
realized e.g., if two individuals have a high number of offspring. In reality
often only a few elements of the set Sx,yi will be realized.
In Eq. (), the set Sx,yi can only be
determined with knowledge of the genotypes of smxi and
sfyi. Nevertheless, in the following the single elements of
Si, sxi and syi, will be characterized via a vector of
phenotypes and characteristics, e.g., height, color of the nose, length of
fur, and so on (Eq. ). The reason for this is that we want to
give a definition of the term “breed”. From the introduction it is known
that individuals belonging to a certain breed need to be recognizable by
their phenotypes, e.g., appearance. If the genotype of the male,
smxi, and female, sfyi, individuals are known
(e.g., genotype for height, color of fur, length of fur, and so on) then the
offspring's genotype can in principle be determined. The offspring are the
elements of the set Sx,yi. Armed with the information
regarding whether certain genes are dominant or recessive one can determine
the corresponding phenotypes. However, the environment may significantly
influence the phenotypic expression. The complexity of the relationship
between genotype and phenotype considering environmental influences was
reviewed by studies such as . To find the set of
offspring Sx,yi for two individuals smxi and
sfyi, i.e., to solve Eq. (), may be very
complicated. This depends on how accurately the phenotypes are specified.
The phenotype “color” can be given by an RGB value, “length” and
“height” in centimeters, “weight” in kilograms, and so on. An element of
the species Si is then characterized by
sxi=(height/cm=35,color of nose/[RGB]=(0.3,0.3,0.8),length of fur/cm=3,…,sex=male)
If one defines that height is always given in centimeters and is always the
first vector element, one can write
sxi=35,(0.3,0.3,0.8),3,…,male=sxi[1],sxi[2],sxi[3],…
In Eqs. () and () the last vector element
specifies the sex. In cases where the individual sxi is a fertile male or
fertile female this entry is “male” or “female”, respectively. Otherwise,
the entry is “infertile”. For the sake of simplicity, in this paper the
notations smxi, sfyi, and sizi are
used. smxi can be considered to be sxi, with “male” being
the last vector entry.
The application of the operator “•” on the fertile male element
smxi and fertile female element sfyi results in a
set which contains doublets of the offspring, i.e., the elements of
Sx,yi, and the realization
probability:
smxi•sfyi=Pmx,yi∪Pfx,yi∪Pix,yi=Px,yiPx,yi=sx,y,1i,px,y,1i,sx,y,2i,px,y,2i,sx,y,3i,px,y,3i…,sx,y,ai,px,y,ai,…,sx,y,wii,px,y,wii
The element sx,y,1i, is realized with the probability of px,y,1i
and so on. The corresponding probability values can be determined by means of
(epi)genetics plus the consideration of the environmental influence, if the
environment has an influence on the genetic expression. In Sect. 4, how to
solve Eqs. () and () is
demonstrated by way of a simple example.