High nonassociativity in order 8 and an associative index estimate

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10400368" target="_blank" >RIV/00216208:11320/19:10400368 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9REwQm9~hE" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=9REwQm9~hE</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1002/jcd.21632" target="_blank" >10.1002/jcd.21632</a>

Result language
angličtina
Original language name
High nonassociativity in order 8 and an associative index estimate
Original language description
Let Q be a quasigroup. Put a(Q) = vertical bar{(x, y, z) is an element of Q(3); x(yz)) = (xy)z}vertical bar and assume that vertical bar Q vertical bar = n. Let delta(L) and delta(R) be the number of left and right translations of Q that are fixed point free. Put delta(Q) = delta(L) + delta(R). Denote by i(Q) the number of idempotents of Q. It is shown that a(Q) >= 2n - i(Q) + delta(Q). Call Q extremely nonassociative if a(Q) = 2n - i(Q). The paper reports what seems to be the first known example of such a quasigroup, with n = 8, a(Q) = 16, and i(Q) = 0. It also provides supporting theory for a search that verified a(Q) >= 16 for all quasigroups of order 8.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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