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Universal Algebra

In: Further Algebra and Applications

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  • P. M. Cohn

    (University College London, Department of Mathematics)

Abstract

Most algebraic systems such as groups, vector spaces, rings, lattices etc. can be regarded from a common point of view as sets with operations defined on them, subject to certain laws. This is done in Section 1.1 and it allows many basic results, such as the isomorphism theorems, to be stated and proved quite generally, as we shall see in Section 1.2. Of the general theory of universal algebra (by now quite extensive), we shall need very little, this forms the subject of Section 1.3; in addition to the basic concepts we define the notion of an algebraic variety, i.e. a class of algebraic systems defined by identical relations, or laws. But there are one or two other topics, not strictly part of the subject that are needed: the diamond lemma forms the subject of Section 1.4, while dependence relations have already been discussed in BA (Section 11.1). There is also the ultraproduct theorem in Section 1.5, a result from logic with many uses (see Chapter 7). The chapter ends in Section 1.6 with an axiomatic development of the natural numbers, regarded as an algebraic system, in an account following Leon Henkin [1960] (see also Cohn (1981)).

Suggested Citation

  • P. M. Cohn, 2003. "Universal Algebra," Springer Books, in: Further Algebra and Applications, chapter 1, pages 1-31, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4471-0039-3_1
    DOI: 10.1007/978-1-4471-0039-3_1
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