September 2011
Beginner
320 pages
8h 17m
English
Content preview from Universal Algebra
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Chapter 1
Algebras
This first, short, chapter sets the stage for the entire text. Many of the fundamental notions and several central examples are introduced. This will allow us to provide examples and motivation for the deeper ideas we encounter in the later chapters.
1.1 Operations
Let A be a set and n a positive integer. We define An to be the set of all n-tuples of elements of A0, and A0 = {∅}.
Remark. More generally, for any two sets A and B, AB denotes the set of all functions from B to A. How do you reconcile this with the definition of An just given? In particular, what about A0? What does the definition tell us about the case A = ∅?
For any A and n as above, we call a function An → A an n-ary operation on A. The natural number ...
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