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Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model. Elementary substructures and elementary extensions
Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are
For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
Two metrics and on X are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants and such that, for every ,, (,) (,) (,).In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different ...
Instead of "0" and "1" one may use, say, the two possible orientations of the plane R 2 ("clockwise" and "counterclockwise"). It is difficult to choose a canonical rule in this case. It is difficult to choose a canonical rule in this case.
Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space. The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form.
Let v 0 equal r 0. Then let v 1 equal r α 1, where α 1 is least such that r α 1 − v 0 is not a rational number. Continue; at each step use the least real from the r sequence that does not have a rational difference with any element thus far constructed in the v sequence. Continue until all the reals in the r sequence are exhausted.
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product).