Chapter 3
The Nuts and Bolts of Universal Algebra
Nuts and bolts are among the tools we use to assemble the objects that dominate our lives. In this chapter we develop tools that allow us to understand how algebras and varieties are put together. Often, it is not assembly that is important, but disassembly—that is, recognizing the component parts of an object under study.
An understanding of the complex interplay of subalgebras, homomorphisms and products is essential to the mastery of universal algebra. In Section 3.4 we illustrate many of these ideas with several concrete examples.
3.1 The isomorphism theorems
Let A and B be sets, and f : A → B a function. Then f induces two maps on the subsets of A and B as follows.
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