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Elements of Programming by Paul McJones, Alexander Stepanov

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Chapter 5

Ordered Algebraic Structures

This chapter presents a hierarchy of concepts from abstract algebra, starting with semigroups and ending with rings and modules. We then combine algebraic concepts with the notion of total ordering. When ordered algebraic structures are Archimedean, we can define an efficient algorithm for finding quotient and remainder. Quotient and remainder in turn lead to a generalized version of Euclid’s algorithm for the greatest common divisor. We briefly treat concept-related logical notions, such as consistency and independence. We conclude with a discussion of computer integer arithmetic.

5.1 Basic Algebraic Structures

An element is called an identity element of a binary operation if, when combined with any other ...

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