An Invitation to General Algebra and Universal Constructions (Universitext)

An Invitation to General Algebra and Universal Constructions (Universitext)

Language: English

Pages: 572

ISBN: 3319114778

Format: PDF / Kindle (mobi) / ePub

Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.

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denominators relatively prime to 10. (i)Determine all nonzero ideals and the structures of the factor-rings Show that each of these factor-rings is isomorphic to a ring Writing them in this way, sketch the diagram of the inverse system of these factor-rings and the canonical maps among them. (ii)Show that the inverse system constitutes a downward cofinal subsystem of the above inverse system. Hence by Exercise 8.5:1 the inverse limits of these two systems are isomorphic, and we shall denote

written as a direct limit in C of a system of objects and maps from D indexed by P ”.) 8.6 Limits and colimits The universal properties defining direct and inverse limits are similar to those defining several other constructions we have seen. Let us recall these. Given two objects X 1, X 2 of a category C, a product of X 1 and X 2 in C is an object P given with morphisms p 1 and p 2 into X 1 and X 2, and universal for this property. Given a pair of parallel morphisms in C, an equalizer of

equalizer of two maps a, b : P → P′. Since a and b are to be morphisms into the direct product object P′, they may be defined by specifying their composites with the projection morphisms p′ X, Y, f  : P′ → F(Y ). Define them so that If L is the equalizer of a and b, and k : L → P the canonical morphism, we see that the universal property of L as an equalizer is equivalent to the statement that the morphisms p X  k : L → F(X) form commuting triangles with the morphisms F(f) and are universal for

identity morphism of R, where in each diagram of (10.6.2), the parenthesized pair shown above the second arrow is an abbreviation for the morphism obtained from its two entries via the universal property of the coproduct Let us now specialize to the case Then the initial object I is the trivial monoid {e}; hence the homomorphism e can only be the map taking every element of R to e. (Contrast this with the case of discussed in � 10.1, where e had for codomain the initial object of a nonzero

H. That is, unless the contrary is mentioned, “maps” between mathematical objects mean maps between their underlying sets which respect their structure. Note that if we wish to refer to a set map not assumed to respect the group operations, we can call this “a map from | G | to | H | ”. The use of letters (μ and for the operations of a group, and the functional notation μ(x, y),  which this entails, are desirable for stating results in a form which will generalize to a wide class of other sorts

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