Nuclear Identification of Some New Loop Identities of Length Five

Abstract

In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Dr\'{a}pal and Jedli\u{c}ka. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Dr\'{a}pal and Jedli\u{c}ka nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws $\{Q_i\}_{i=1}^{12}$ induced by our nuclear identification form four cycles in the following sequential format: $\big(Q_{4i-j}\big)_{i=1}^3,~j=0,1,2,3,$ and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities $\{Q_i^*\}_{i=1}^{12}$ induced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws $\{Q_i\}_{i=1}^{12}$ were found.