Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same
  Variety   The isotopic invariance or universality of types and varieties of quasigroups
and loops described by one or more equivalent identities has been of interest
to researchers in loop theory in the recent past. A variety of
quasigroups(loops) that are not universal have been found to be isotopic
invariant relative to a special type of isotopism or the other. Presently,
there are two outstanding open problems on universality of loops: semi
automorphic inverse property loops(1999) and Osborn loops(2005). Smarandache
isotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002.
But in this work, the concept is re-restructured in order to make it more
explorable. As a result of this, the theory of Smarandache isotopy inherits the
open problems as highlighted above for isotopy. In this short note, the
question 'Under what type of S-isotopism will a pair of S-quasigroups(S-loops)
form any variety?' is answered by presenting a pair of specially S-isotopic
S-quasigroups(loops) that both belong to the same variety of
S-quasigroups(S-loops). This is important because pairs of specially S-isotopic
S-quasigroups(e.g Smarandache cross inverse property quasigroups) that are of
the same variety are useful for applications(e.g cryptography).

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The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph