A NOTE ON THE AUSTIN'S GROUPOIDS
Jung-R. Cho, Jo Dudek · East Asian Mathematical Journal · 2006
Abstract. On a groupoid satisfying the Austin’s identity, everyn-ary linear term is essentially n-ary. That is, if a term has novariables appearing more than once, then the term depends onevery variable it involves. 1. IntroductionA groupoid is a pair (G,·) of a set G and a binary operation ‘·’defined on G. A term or a word in a set X = {x 1 ,x 2 ,···} of symbolsis an expression built up from X using the groupoid operation. Weuse the notation x 2 for the term xx. Thus x 2 x, xx 2 and x 2 x 2 represent(xx)x, x(xx) and (xx)(xx), respectively.A term is called n-ary if it involves n distinct variables in its expres-sion, and linear if each variable appears at most once in the expression.On a groupoid (G,·), an n-ary term f(x 1 ,x 2 ,··· ,x n ) defines a map-ping of G n into G by substitution. A mapping defined by a term inthis way is called a term function. An n-ary term is called essentiallyn-ary over a groupoid (G,·) if, as a term function, it depends on eachx i for i = 1,2,··· ,n. That is,f(a