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And if each linearization is only surjective, and a family of right inverses is smooth tame, then P is locally surjective with a smooth tame right inverse.
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The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other, i.e. LGL = L and GLG = G and one uniquely determines the other. They are not left or right inverses of each other however.
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Left and right inverses are not necessarily the same. If is a left inverse for , then may or may not be a right inverse for ; and if is a right inverse for , then is not necessarily a left inverse for . For example, let denote the squaring map, such that for all in , and let denote the square root map, such that for all . Then for all in ; that is, is a right inverse to .
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Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on X, while the least element is the indiscrete topology.
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Algebraic structures between magmas and groups. A loop is a quasigroup with an identity element; that is, an element, e, such that :x ∗ e = x and e ∗ x = x for all x in Q. It follows that the identity element, e, is unique, and that every element of Q has unique left and right inverses (which need not be the same). A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, , taking its subtraction operation as quasigroup multiplication yields a pique with the group identity (zero) turned into a "pointed idempotent". (That is, there is a principal isotopy .) A loop that is associative is a group.
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