Ultraproduct
المؤلف:
Bell, J. L. and Slomson, A. B.
المصدر:
Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.
الجزء والصفحة:
...
13-2-2022
995
Ultraproduct
Let 
be a language of first-order predicate logic, let 
be an indexing set, and for each 
, let 
be a structure of the language 
. Let 
be an ultrafilter in the power set Boolean algebra 
. Then the ultraproduct of the family 
is the structure 
that is given by the following:
1. For each fundamental constant 
of the language 
, the value of 
is the equivalence class of the tuple 
, modulo the ultrafilter 
.
2. For each 
-ary fundamental relation 
of the language 
, the value of 
is given as follows: The tuple ![([x_1]_u,...,[x_n]_u)](https://mathworld.wolfram.com/images/equations/Ultraproduct/Inline19.svg)
is in 
if and only if the set ![<span style=]()
{i in I|(x_1(i),...,x_n(i))}" src="https://mathworld.wolfram.com/images/equations/Ultraproduct/Inline21.svg" style="height:22px; width:189px" /> is a member of the ultrafilter 
.
3. For each 
-ary fundamental operation 
of the language 
, and for each 
-tuple ![([x_1]_u,...,[x_n]_u)](https://mathworld.wolfram.com/images/equations/Ultraproduct/Inline27.svg)
, the value of ![f^((A))([x_1]_u,...,[x_n]_u)](https://mathworld.wolfram.com/images/equations/Ultraproduct/Inline28.svg)
is ![[f^((A))(x_1,...,x_n)]_u](https://mathworld.wolfram.com/images/equations/Ultraproduct/Inline29.svg)
.
The ultraproduct 
of the family 
is typically denoted 
.
REFERENCES
Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.
Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981.
http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.
Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.
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