zoology - Transitivity of Species Definitions

There are many definitions of "species" which usually take the form

Two individuals are of the same species if ...

An implied (rarely made explicit) property of any sensible species definition is that the relation "is the same species as" should be an equivalence relation (the reason why this is sensible is that only such a relation partitions the set of all individuals into equivalence classes, that is, separate species). In particular, it should be transitive, that is

If A and B are of the same species, and B and C are of the same
species, then it follows that A and C are also of the same species.

It appears to me that all species definitions in use violate that property.

Are the Chihuahua and the Great Dane of the same species?

Not according to the "biological" species definition, which states:

Two individuals are of the same species if and only if they can produce fertile offspring.

Since that is (presumably) not true of Chihuahua and Great Dane, according to that definition, the two dog breeds are not of the same species.

However...

Let us say that the Chihuahua can produce fertile offspring with the Dachshund...

... and the Dachshund with the Golden Retriever...

... and that at last with the Great Dane as desired.

Then, if we assume transitivity, Chihuahua and Great Dane are of the same species - contradiction!

Are the Chimpanzee and the White Oak of the same species?

Intuitively, they are of course not, the very idea seems ridiculous. Sure enough, with the definition (a variation of the "genetic" species definition)

Two individuals are of the same species if and only if their genome differs in less than 0.5% of base pairs.

the two are indeed not of the same species.

However, we can again construct a "chain" of intermediate individuals, each of which is similar enough to its predecessor to satisfy the definition, eventually connecting Ape and Tree and violating transitivity again.