In mathematical
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
as ''x'', the restriction of
to
is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
on
. Here
denotes the set of strictly increasing sequences of members of
, or equivalently the family of subsets of
with
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for
Bjarni Jónsson
Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory. He was emeritus distinguished professor of mathematics at Vanderbi ...
.
showed that for every ordinal λ there is an ω-Jónsson function for λ.
Kunen's proof of
Kunen's inconsistency theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.
Some consequences of Kunen's theorem (or its proof) are:
*There is no ...
uses a Jónsson function for
cardinals
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of cardinal in the family Cardinalidae
**''Cardinalis cardinalis'', or northern cardinal, th ...
λ such that 2
λ = λ
ℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. 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 algebra Jónsson is a surname of Icelandic origin, meaning ''son of Jón''. In Icelandic names, the name is not strictly a surname, but a patronymic. The name refers to:
*Arnar Jónsson (actor) (born 1943), Icelandic actor
*Arnar Jónsson (basketball) (born ...
s exists in any cardinality, so there are no infinitary analogues of
Jónsson cardinal In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.
An uncountable cardinal number κ is said to be ''Jónsson'' if for every function ''f'': �sup><ω → κ there is a set ''H'' of ...
s.
References
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{{DEFAULTSORT:Jonsson Function
Set theory
Functions and mappings