Abstract
This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns. The generating function of the said patterns has been reported earlier by the author [1] [2]. The paper describes how these non-associative and non commutative properties can be established by using the Cayley table on which a binary operation is defined to act on the 123-avoiding and 132-avoiding patterns of Aunu permutations using a pairing scheme. Our results have generated larger matrices from permutations of points of the Aunu patterns of prime cardinality. It follows that the generated symbols can be used in further studies and analysis in cryptography and game theory thereby providing an interdisciplinary approach and applications of these important permutation patterns.
Highlights
Non-associative algebraic structures arise in many situations
(i + j th where i enumerates the cycles formed in permutations of elements of Ī© and j represents the shift in permutation of elements of the symbols of Ī© in a pairing scheme defined by (4)
This has clearly shown to us that, the above constructed structures using Cayley table are non-associative and non-commutative. It follows that the permuted structures of Aunu scheme give rise to non-associative structures where points in
Summary
Non-associative algebraic structures arise in many situations. Cayley octonions are a notorious example, but there are far more; for example, nonassociative loop arise in cordinatization of projective planes and the Einstein velocity addition in relativity theory forms a nonassociative loop. Self distributive algebras appear naturally in the study of Braids [3]. The 123-avoiding class of the Aunu permutation patterns which have been found to be of both combinatorics and group theoretic importance [1] can be used to construct some structures which are non-associative as well as non-commutative using Cayley table with a binary operation defined to act on such patterns. Non-associative structures include structures like groupoids, quasigroup and loops, nonassociative semi-rings. (2016) Non-Associative Property of 123-Avoiding Class of Aunu.
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