Further, the definition of the frequency square in the language of multisets suggests
First, define the functions induced by left multiplication by via , and the functions induced by right multiplication by via . The condition of being a row-quasigroup is then that is a bijection for all , etc. However, instead we now define the multi-image of a function as the multiset over where the multiplicity of is the number of solutions to .
In a quasigroup, all of these multiplicities are by definition 1 (and in particular the image is the entire quasigroup), but in our generalization we simply require the identity for all . I am sure this generalization has been considered many times before, but I know of no established name for it, so we will call it a permutation square.
A similar but weaker condition is that the induced multisets are isomorphic, which is to say that the images may not coincide, but the multiplicities are in one-to-one correspondence. Explicitly, we need the existence, for some arbitrarily chosen : a bijection , for any , such that the multiplicity of in is the same as the multiplicity of in ; and a bijection , for any , such that the multiplicity of in is the same as the multiplicity of in . That’s rather complicated, but the intuitive version is easy enough to understand: the rows and columns have the same frequency distributions of elements. This we will call a frequency square, although I believe the extant usage of the term is slightly different.
I am pretty sure that we can axiomatize frequency squares so that they form a quasivariety:
implies
implies
The above can be replaced with implying if we don’t require the left and right multisets to coincide.
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There are three ways to strengthen a frequency square without getting to a permutation square (or at least three that we consider).
First, a semipermutation square is a frequency square wherein every isomorphism is an involution - and since the composition of involutions is not necessarily an involution, this requires that we enforce the existence of bijections for every element and not just a single arbitrarily chosen one. Since the identity is clearly an involution, a permutation square is a semipermutation square.
Second, a strong frequency square is a frequency square that preserves the indices of induced functions, i.e. . This, like the above, requires explicitly imposed bijections for every element, and then the property can be expressed as an identity .
Third, a coherent frequency square is a frequency square where the same pre-images remain in the same frequency class, i.e., if , then , for all . This condition has also surely been considered before, and in fact I believe it is actually equivalent to the intersection of left and right simplicity in semigroups?
These notions inspire a strengthening of frequency square that is actually incomparable with quasigroups: that of a function square, where any pair of induced functions are isomorphic. To wit, we need the existence, for some arbitrarily chosen : a bijection , for any , such that ; and a bijection , for any , such that . Note that the concepts of strength and coherence apply equally well, and there is even an analogue of semipermutation square available, which we shall term semiconstant magma.
In any case, a quasigroup is clearly a permutation square, and while a permutation square is not necessarily strong or coherent, a quasigroup is in a trivial way, since every element has multiplicity one, any bijection will suffice for its multiset isomorphisms.
There is a lot to explore here.