Antiassociative magmas

Abstract

A magma S is said to be antiassociative if $$(ab)c \ne a(bc)$$ ( a b ) c ≠ a ( b c ) holds for any elements a, b, c of S. Antiassociative magmas lie at the opposite pole from associative ones, known as semigroups. In this paper, we study antiassociative magmas focusing on their properties and examples that can be seen from Cayley tables. We provide a test for the antiassociativity of a finite magma and give some general methods for constructing antiassociative magmas. We also characterize, describe, and count all antiassociative magma structures on a 3-element set, all their isomorphism classes, and all classes of equivalent magmas of this type.