Erdős, Pál; Hajnal, András On a problem of B. Jonsson. (English) Zbl 0171.26501 Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 19-23 (1966). The problem of B. Jónsson which is dealt with is the following one: Does there exist an algebra of power \(\alpha\) with no proper subalgebra of the same power? (An algebra has finitely many finitary operations.) The authors investigate also another problem: for which cardinals \(\alpha\) is there an algebra without infinite independent subsets? An affirmative answer implies an affirmative answer to Jónsson’s problem. Results: The generalized continuum hypothesis implies that the answer to Jónsson’s probles is ”yes” for \(\alpha\) non limit. The answer is ”yes” for the second problem (thus also for Jónsson’s one) for \(\omega_n,n < \omega_0\). The answer is ”yes” for Jónsson’s problem for \(\alpha\) measurable. For any \(\alpha\), there is an algebra with one \(\omega\)-ary operation without a proper subalgebra. Almost all results proved here are consequences of the results of P.Erdős, A.Hajnal and R.Rado [Acta Math. Acad. Sci. Hung. 16, 93-196 (1965; Zbl 0158.26603)]. Reviewer: L.Bukovský Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 17 Documents MSC: 03E50 Continuum hypothesis and Martin’s axiom 08A65 Infinitary algebras Keywords:set theory Citations:Zbl 0158.26603 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{A. Hajnal}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 19--23 (1966; Zbl 0171.26501)