Enlargement
المؤلف:
Gehrke, M.; Kaiser, K.; and Insall, M.
المصدر:
"Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36
الجزء والصفحة:
...
13-2-2022
2393
Enlargement
In geometry, the term "enlargement" is a synonym for expansion.
In nonstandard analysis, let 
be a set of urelements, and let 
be the superstructure with individuals in 
:
1. 
,
2. 
,
3. 
.
Let 
be a superstructure monomorphism, with 
and 
for 
. Then 
is an enlargement of 
provided that for each set 
in 
, there is a hyperfinite set 
that contains all the standard entities of 
.
It is the case that 
is an enlargement of 
if and only if every concurrent binary relation 
satisfies the following: There is an element 
of the range of 
such that for every 
in the domain of 
, the pair 
is in the relation 
.
REFERENCES
Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.
Gonshor, H. "Enlargements Contain Various Kinds of Completions". In Proc. 1972 Victoria Symposium on Nonstandard Analysis. New York: Springer-Verlag, pp. 60-70, 1974.
Gonshor, H. "Enlargements of Boolean Algebras and Stone Spaces". Fund. Math. 100, 35-59, 1978.
Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.I
nsall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.I
nsall, M. "Geometric Conditions for Local Finiteness of a Lattice of Convex Sets." Math. Moravica 1, 35-40, 1997.
Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra." Zeitschr. f. Math., Logik, und Grundlagen d. Math. 37, 525-532, 1991.
Luxemburg, W. A. J. Applications of Model Theory to Algebra, Analysis, and Probability. New York: Holt, Rinehart, and Winston, 1969.
Robinson, A. Nonstandard Analysis. Amsterdam, Netherlands: North-Holland, 1966.Robinson, A. "Germs." In Applications of Model Theory to Algebra, Analysis and Probability (International Sympos., Pasadena, Calif., 1967).
Schmid, J. "Completing Boolean Algebras by Nonstandard Methods." Zeitschr. für Math. Logik u. Grundlagen der Mathematik 20, 47-48, 1974.
Schmid, J. "Nonstandard Constructions for Join-Extensions of Lattices." Houston J. Math. 3, 423-439, 1977.
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