English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The isotopic invariance or universality of types and varieties of quasigroupsand loops described by one or more equivalent identities has been of interestto researchers in loop theory in the recent past. A variety ofquasigroups(loops) that are not universal have been found to be isotopicinvariant relative to a special type of isotopism or the other. Presently,there are two outstanding open problems on universality of loops: semiautomorphic inverse property loops(1999) and Osborn loops(2005). Smarandacheisotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002.But in this work, the concept is re-restructured in order to make it moreexplorable. As a result of this, the theory of Smarandache isotopy inherits theopen problems as highlighted above for isotopy. In this short note, thequestion '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-isotopicS-quasigroups(loops) that both belong to the same variety ofS-quasigroups(S-loops). This is important because pairs of specially S-isotopicS-quasigroups(e.g Smarandache cross inverse property quasigroups) that are ofthe same variety are useful for applications(e.g cryptography).

A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety