244 ON CONVEX LINEAR FORMS 1) SIEMION FAJTLOWICZ and JAN MYCIELSKI 1. R denotes the set of real numbers and 91p=(R, px+(1-p)y), where p~R. We shall use also the notation xoy=px+(1-p)y. It is visible that the algebra 9.[p satisfies the following two equations xox-'-x, (1) (x oy)o (uo v)-"- (xo u) o (yo v). (2) Let T~ be the equational theory (see [1]) of the algebra 9.Ip, i.e., the set of all equations in o, universally true in 9/p. It is the purpose of this paper to prove the following theorem. THEOREM. (1) and (2) constitute an axiomatisation of Tp if and only if p is a transcendental number. The 'if' was announced in [2]. The 'only if', which we got later, uses a lemma whose proof we owe to E. G. Straus. Our theorem suggests the following problems which remain open. Is Tp finitely based (i.e., finitely axiomatisable by equations in o) for every peR? How to extend this Theorem to other convex linear forms PlXl +"" +p,x,, where pl +... +p,= 1 ? For other problems see section 5. 2. To prove our Theorem we need some notations and lemmas. Let V= {xl, Xz.... } be the set of all variables of Tp and S the set of all terms (i.e., polynomial expressions in o and V) of Tp. LEMMA 1. l f p is transcendental then T~=("IT ~. q~R Proof. Every term t~S defines a function in 9.Iq (q~R) which is a convex linear form in the variables of t and the coefficients of this form are polynomials in q with integral coefficients. Thus an equation in o is true in 9~ if and only ifq satisfies a certain set of non trivial identities P(q)=0, where P are some polynomials with integral coefficients. For some equations this set may be empty. If q is transcendental then only such equations are true in 9/q for which the corresponding set is empty. Q.E.D. 1) This work was supported by NSF Grant GP-19405. Presented by R. McKenzie. ReceivedNovember 14, 1973. Accepted for publication in final form March 11, 1974.