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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 171.26501

**Autor: ** Erdös, Pál; Hajnal, András

**Title: ** On a problem of B. Jonsson (In English)

**Source: ** Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 19-23 (1966).

**Review: ** 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.Erdös, A.Hajnal* and *R.Rado* [Acta Math. Acad. Sci. Hung. 16, 93-196 (1965; Zbl 158.26603)].

**Reviewer: ** L.Bukovský

**Classif.: ** * 03E50 Continuum hypothesis and generalizations (logic)

08A65 Infinitary algebras

04A30 Continuum hypothesis and generalizations

**Index Words: ** set theory

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