MoReTax

Combined relationships

Call R the set of all basic relationships. R = {º, Ì, É, Å, !}. Every subset S of R is a combined relationship. This means that if PT1 has a relationship S to PT2 then one of the relationships that belong to S is the basic relationship between PT1 and PT2.
(PT1 S PT2) Þ $ Ri Î S | (PT1 Ri PT2)

There are 32 (2⁵) different combined relationships:
Æ, {º}, {Ì}, {É}, {Å}, {!}, {º, Ì}, {º,  É}, {º, Å}, {º, !}, {Ì, É}, {Ì, Å}, {Ì, !}, 
{É, Å}, {É, !}, {Å, !}, {º, Ì, É},  {º, Ì, Å}, {º, Ì, !}, {º, É, Å}, {º, É, !}, {º, Å, !}, 
{Ì, É, Å},  {Ì, É, !}, {Ì, Å, !}, {É, Å, !}, {º, Ì, É, Å}, {º, Ì, É, !}, {º, Ì, Å, !}, 
{º, É, Å, !}, {Ì, É, Å, !} and {º, Ì, É, Å, !}.

Note that it would be an error to treat the empty relationship (Æ) as being equal to R5 (!). The empty relationship is only used for the representation of a combination of contradictory opinions about relationships.
The relationship R (i.e. the set of all basic relationships) describes the fact that nothing is known about the real relationship between two taxonomic concepts.

The concatenation operator

What happens if there is a relationship S1 between PT1 and PT2 and another relationship S2 between PT2 and PT3?

For this purpose an operator “®“ between two combined relationships can be defined, the result of which is another combined relationship:

S1 ® S2 = S3 if and only if S3 is the relationship between PT1 and PT3 that can be deduced from the fact that S1 is the relationship between PT1 and PT2 and S2 the relationship between PT2 and PT3.

The results from this operator can be described in a 32 x 32 table.

Examples:
{º} ® S = S for every S
{º, Ì} ® {Ì} = {Ì}
{É} ® {É} = {É} but
{Ì}® {Å} = {Ì, Å, !} and
{Å} ® {Å} = {º, Ì, É, Å, !}.

In the example mentioned above about  Racomitrium affine (F. Weber & D. Mohr) Lindb. it was asserted that
      PT1 Ì PT4 and PT1 É PT6. 
Therefore 
      PT4 É PT1 and PT6 Ì PT1. 
Now we can ask about the relationships between PT4 and PT8 on the one side and between PT6 and PT8 on the other side. Both of them follow from the already known relationships.

     If PT4 S1 PT1, PT6 S2 PT1 and PT1 S3 PT8 so are S1 = {É}, S2 = {Ì} and S3 = {Å}. 

If the relationship which arises through concatenation between PT4 and PT8 is called  S4  (and S5 the one between PT6 and PT8), so 
     S4 = S1 ® S3 (S4 = {É}® {Å}) 
which means 
     S4 = {É, Å} 
or expressed differently that 
    PT4 {É, Å} PT8
and 
     S5 = S2 ® S3 (S5 = {Ì}® {Å}) 
which means 
     S5 = {Ì, Å, !} 
or in other words 
    PT6 {Ì, Å, !} PT8.

If we take into account that according to KOPERSKI et al. (2000) 
“R. heterostichum var. gracilescens is to be included in the Synonymy of R. sudeticum“, i.e.  PT1 Å PT8 it is possible to say that both PT6 and PT8 include the type R. heterostichum var. gracilescens and therefore CANNOT exclude each other.

It can be asserted therefore that actually PT6 {Ì , Å} PT8 but that PT4 {É, Å} PT8 remains. It can also be deduced that PT5 {Ì , Å, !} PT8.

BACK TO TOP

Information transmittability

If we have several taxonomic concepts with different relationships among them, then new relationships arise through concatenation. How can these final relationships be interpreted so the user can get an indication about the quality of the information transmission?

Four options exist for the transmittability to the taxonomic concept PT2 of information linked to the taxonomic concept PT1:

• Fully transmittable if xÎ PT2 Þ xÎ PT1;
  it follows from PT1 S PT2 that S = {º}, S = {É} or S = {º, É}

• Partially transmittable if $xÎ PT2 | xÎ PT1 and $yÎ PT2 | yÏ PT1;
  it follows from PT1 S PT2 that (Ì Î S or Å Î S) but also that ! Ï S

• Dubiously transmittable if it might be that $xÎ PT1 | xÎ PT2;
  it follows from PT1 S PT2 that ! Î S but S ¹ {!}

• Not transmittable if xÎ PT2 Þ xÏ PT1;
  it follows from PT1 S PT2 that S = {!}

Linked information should be transmitted only in the first three cases (with an appropriate comment as to possible caveats).

In the case of the different taxonomic views about Racomitrium affine, if the transmittability  of information from other taxonomic views to PT1 [Racomitrium affine (F. Weber & D. Mohr) Lindb. sec. KOPERSKI & al. (2000)] is to be considered,  it can be said that:

• information  linked to PT2, PT3 and PT4 are fully transmittable, 
• information  linked to PT5, PT6, PT7 and PT8 are partially transmittable and 
• information  linked to PT9 are not transmittable.

If the discussed relationships between PT4, PT5 or PT6 and PT8 are considered, then it can be said that

• information  linked to PT4 and PT6 are partially transmittable to PT8 but
• information  linked toPT5 are dubiously transmittable to PT8. 

BACK TO TOP

Relationship qualifier

If the expert has doubts about the relationship, he can mark it with “?” (doubtful).
So the new expressions º? … !? and S? = {Ri? | Ri Î S} are meaningful.

In the example already mentioned KOPERSKI & al. have qualified in two cases the indicated relationship as assumptions, so that these relationships should be described as 
PT1 Ì? PT4 and PT1 É? PT7.

This marker is always transmitted to the derived relationships when "concatenated".

Therefore the relationship between PT4 and PT8 is S4? = {É?, Å?} and not merely S4 = {É, Å}.

For the interpretation of transmittability it can be asserted that the transmittabilities inherit the "doubtful" qualification in such a way that:

Let PT1 S PT2,
then the transmittability which is associated with S is "doubtful" if and only if
Ri? for every Ri that belongs to S (Ri Î S Þ Ri?). 

In this sense it can be said that factual data which are bound to Racomitrium heterostichum (Hedw.) Brid. sec. SMITH (1980) are not only only "partially" transmittable to Racomitrium sudeticum (Funck) Bruch & Schimp. sec. CORLEY & al. (1981/1991) but even that this partial transmittability is doubtful.

BACK TO TOP

LITERATURE

BERENDSOHN, W. G. (1995): The concept of "potential taxa" in databases. - Taxon 44: 207-212 

KOPERSKI, M., SAUER, M., BRAUN, W. & GRADSTEIN, S. R. 2000: Referenzliste der Moose Deutschlands. Dokumentation unterschiedlicher taxonomischer Auffassungen. Schr.-R. f. Vegetationskunde. 34: 1-519.

ZHONG, Y., JUNG, S., PRAMANIK, S. & BEAMAN, J. H. 1996: Data model and comparison and query methods for interacting classifications in a taxonomic database. Taxon 45: 223-241.