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On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form (a, b], with a, b ∈ R is a ring in the measure-theoretic sense. If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
The ring settings, or Ringstellung, are used to change the position of the alphabet ring relative to the internal wiring. Notch and alphabet ring are fixed together. Changing the ring setting will therefore change the positions of the wiring, relative to the turnover-point and start position. The ring setting will rotate the wiring.
For any commutative ring A and multiplicative set S in A, the localization S −1 A is the zero ring if and only if S contains 0. If A is any ring, then the ring M 0 (A) of 0 × 0 matrices over A is the zero ring. The direct product of an empty collection of rings is the zero ring. The endomorphism ring of the trivial group is the zero ring.
The operations in this ring are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by End R (V). The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a ...
Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object. Morphisms of pointed sets.
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