High nonassociativity in order 8 and an associative index estimate | Zendy

Drápal Aleš | Zendy; Valent Viliam | Zendy

AI Assistant Blog Pricing

Home ZAIA Blog

Premium

High nonassociativity in order 8 and an associative index estimate

Author(s) -

Drápal Aleš,

Valent Viliam

Publication year - 2019

Publication title -

journal of combinatorial designs

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.618

H-Index - 34

eISSN - 1520-6610

pISSN - 1063-8539

DOI - 10.1002/jcd.21632

Subject(s) - quasigroup , mathematics , associative property , order (exchange) , index (typography) , point (geometry) , combinatorics , arithmetic , discrete mathematics , algebra over a field , pure mathematics , computer science , geometry , finance , world wide web , economics

Let Q be a quasigroup. Put a ( Q ) = ∣ { ( x , y , z ) ∈ Q 3 ;x ( y z )= ( x y ) z } ∣ and assume that ∣ Q ∣ = n . Let δ L and δ R be the number of left and right translations of Q that are fixed point free. Put δ ( Q ) = δ L + δ R . Denote by i ( Q ) the number of idempotents of Q . It is shown that a ( Q ) ≥ 2 n − i ( Q ) + δ ( Q ) . Call Q extremely nonassociative if a ( Q ) = 2 n − 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 .

This content is not available in your region!

Continue researching here.

Having issues? You can contact us here

Accelerating Research