Chapter 1 contains the basics of semigroups: definitions, elementary concepts, and fundamental examples. We assume some familiarity with standard notions of point-set topology (see [124, 167]); the algebraic portions of Chap. 1 are, however, completely self-contained. Without going into any details whatsoever, it is perhaps prudent to remark at this point that our main interest centers around asymptotics, invariance questions, etc. Our treatment is a reflection of this. We concentrate on algebraic concepts corresponding to such phenomena as absorption, stability, and invariance: zeros, simple semigroups, minimal ideals, maximal subgroups, and so on. We strive to keep the digressions at a minimum. Clifford and Preston  offers a wealth of information on all aspects of algebraic semigroups, and this text is recommended to any reader interested in a much more elaborate treatment of this fascinating subject.
KeywordsLeft Ideal Simple Semigroup Topological Semigroup Nonnegative Matrice Primitive Idempotent
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