- References
In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function 
with the property that, for any subset y of x with the same cardinality as x, the restriction of 
to 
is surjective on 
. Here 
denotes the set of strictly increasing sequences of members of , or equivalently the family of subsets of with order type 
, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson.
Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. {{harvs|txt|authorlink=Fred Galvin|last=Galvin|last2= Prikry|year=1976}} gave a simple proof for the general case.
The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality. In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals.
References
- {{Citation | last1=Erdős | first1=P. | author1-link=Paul Erdős | last2=Hajnal | first2=András | author2-link=András Hajnal | title=On a problem of B. Jónsson |mr=0209161 | year=1966 | journal=Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques | issn=0001-4117 | volume=14 | pages=19–23}}
- {{Citation | last1=Galvin | first1=Fred | last2=Prikry | first2=Karel | title=Infinitary Jonsson algebras and partition relations | doi=10.1007/BF02485843 |mr=0434822 | year=1976 | journal=Algebra Universalis | issn=0002-5240 | volume=6 | issue=3 | pages=367–376}}
- {{Citation | last1=Jónsson | first1=Bjarni | title=Topics in universal algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0058648 |mr=0345895 | year=1972 | volume=250}}
- {{Citation | last1=Kanamori | first1=Akihiro | title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-3-540-00384-7 | year=2003|page=319}}
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