On a Problem of B . Jónsson

B. Jónsson has asked : Is there an algebra of power a with no proper subalgebra of power a? H. J. Keisler and F. Rowbottom proved that the answer is affirmative for every a, provided Gödel's constructibility axiom holds [1]. The main aim of this paper is to prove, using the generalized continuum hypothesis (G .C .H. in what follows), that the answer is affirmative for each non limit cardinal (see Theorem 1). This result can be extended for locally finite algebras, too (see Theorem 3). Without using G .C .H. we prove that the answer to the problem is "yes", provided a to ~ (,) (see Theorem 2). We also prove without using G .C .H. that for every a there is an algebra of power a with one wary operation which has no proper subalgebra of power a. (See Theorem 5). Finally, we state another problem and some results relevant to Jónsson's problem. Then for each infinite cardinal a there exists an algebra of pourer a+ witq one binary operation such that it has no proper sebalgebra of power a+. THEOREM 2. For each finite n there exists an algebra of power (o,, with one binary operation which has no proper subalgebra of power (o,,. Proof of Theorem 1. Put A = a+. By Theorem 17 of [2] there exists a function f E°+ > °+ u+ satisfying the following condition (1) If B c u+, IBS = a+, then f (BX B) = a+. The algebra obviously satisfies the requirement of Theorem 1. Note that if J3 is a singular limit cardinal and the G .C .H. holds, there is no function f satisfying (1) with 13 instead of a+ (See Theorem 20 of [2]). We do not know if (1) can be satisfied if we replace a+ by a strongly incompact inaccessible cardinal (3. Proof of Theorem 2. We prove the weaker statement, namely that there exists an algebra <A, f with one n { 1-ary operation satisfying the requirement .