Enabling conditions for interpolated rings
نویسنده
چکیده
modulo the condition that ' (a) = a for all a 2 A if P holds. If ' is the injection map A B, then C is simply the union A [ fb 2 B : Pg, which is a subring of B. This is the case of most interest when dealing with discrete rings. Interpolated rings are used to construct Brouwerian examples. For example, the interpolated rings in Z Q provide a Brouwerian example of a discrete ring C and a nitely generated ideal I of C for which the assertion 1 2 I or 1 = 2 Icannot be justi ed. If, in this example, P states that a certain binary sequence contains a 1, then C is countable. Of course we need not restrict ourselves to rings. The same idea applies to other mathematical structures as well. Jesper Carlström pointed out that the P -interpolation C of the map ' : f0; 1g ! f0g, together with the natural map from f0; 1g toC is the Goodman-Myhill example showing that the axiom of choice implies the law of excluded middle [1].
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