Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pentagonal quasigroups, their translatability and parastrophes   Any pentagonal quasigroup is proved to have the product xy = R(x)+y-R(y)
where (Q,+) is an Abelian group, R is its regular automorphism satisfying
R^4-R^3+R^2-R+1 = 0 and 1 is the identity mapping. All abelian groups of order
n<100 inducing pentagonal quasigroups are determined. The variety of
commutative, idempotent, medial groupoids satisfying the pentagonal identity
(xy*x)y*x = y is proved to be the variety of commutative pentagonal
quasigroups, whose spectrum is {11^n : n = 0,1,2,...}. We prove that the only
translatable commutative pentagonal quasigroup is xy = (6x+6x)(mod11). The
parastrophes of a pentagonal quasigroup are classified according to well-known
types of idempotent translatable quasigroups. The translatability of a
pentagonal quasigroup induced by the additive group Zn of integers modulo n and
its automorphism R(x) = ax is proved to determine the value of a and the
possible values of n.

36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat