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Transfer theorems for \(\pi\)-monads. (English) Zbl 0686.03038

The purpose of this paper is to give a brief introduction to the Bennighofen-Richter theory of \(\pi\)-monads [B. Benninghofen and M. M. Richter, Fundam. Math. 128, 199-215 (1987; Zbl 0633.03067)] and to present a simple version of some transfer principles for \(\pi\)- monads. Let V(S) be a superstructure, \(A\in V(S)\) and \(U\subset P(A)\). The set \(\mu =\cap^*_{w\in U}W\) is called a monad on \({}^*A\) generated by the set U. A theorem is proved which shows how new monads may be obtained from given ones by means of increasing set functions. Let \((\mu_ i)_{i\in I}\) be a family of monads on the sets \((^*A_ i)_{i\in I}\) given by filters (\({\mathcal F}_ i)_{i\in I}\). Let \({\mathcal F}=\prod_{i\in I}{\mathcal F}_ i\). For \(j\in^*I\) the set \(\pi_{\mu_ j}=\cap^*_{F\in {\mathcal F}}F_ j\) is called a \(\pi\)-monad. The author proves the following “monotone transfer theorem”. Let \((\mu_ i)_{i\in I}\) be a family of monads on the sets \((^*A_ i)_{i\in I}\); suppose that \(\phi\) (x,y) is a formula and \((B_ i)_{i\in I}\) is a standard family such that \(\phi (x,{\mathbb{B}}_ i)_{i\in I}\) is monotone (here a boldface variable denotes a finite sequence of variables). Then the following are equivalent: (a) \(\phi (\mu_ i,^*{\mathbb{B}}_ i)\) for all \(i\in I\), (b) \(\phi (\pi_{\mu_ j},^*{\mathbb{B}}_ j)\) for all \(j\in^*I\).
Reviewer: G.Pestov

MSC:

03H05 Nonstandard models in mathematics

Citations:

Zbl 0633.03067
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References:

[1] Aloeverio, S.; Fenstad, J. E.; Hoegh-Krohn, R.; Lindstrom, T., Nonstandard Methods in Stochastic Analysis and Mathematical Physics (1986), Academic Press: Academic Press New York
[2] Benninghofen, B.; Richter, M. M., General theory of superinfinitesimals, Fund. Math., 128, 199-215 (1987) · Zbl 0633.03067
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