Abstract
This chapter contains those elements of the structure theory of finite monoids that we shall need for the remaining chapters. It also establishes some notation that will be used throughout. More detailed sources for finite semigroup theory include [KRT68, Eil76, Lal79, Alm94, RS09]. Introductory books on the algebraic theory of semigroups, in general, are [CP61, CP67, Hig92, How95]. A detailed study of some of the most important transformation monoids can be found in [GM09]. In this book, all semigroups and monoids will be finite except for endomorphism monoids of vector spaces and free monoids. On a first reading, it may be advisable to skip the proofs in this chapter.
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References
J. Almeida, Finite Semigroups and Universal Algebra. Series in Algebra, vol. 3 (World Scientific, River Edge, NJ, 1994). Translated from the 1992 Portuguese original and revised by the author
A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups. Volume I. Mathematical Surveys, No. 7 (American Mathematical Society, Providence, RI, 1961)
A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups. Volume II. Mathematical Surveys, No. 7 (American Mathematical Society, Providence, RI, 1967)
S. Eilenberg, Automata, Languages, and Machines. Volume B (Academic, New York, 1976). Pure and Applied Mathematics, vol. 59. With two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by B. Tilson.
O. Ganyushkin, V. Mazorchuk, Classical Finite Transformation Semigroups, An Introduction. Algebra and Applications, Number 9 (Springer, Berlin, 2009)
J.A. Green, On the structure of semigroups. Ann. of Math. (2) 54, 163–172 (1951)
P.M. Higgins, Techniques of Semigroup Theory. Oxford Science Publications (The Clarendon Press Oxford University Press, New York, 1992). With a foreword by G.B. Preston
J.M. Howie, Fundamentals of Semigroup Theory. Oxford Science Publications, London Mathematical Society Monographs. New Series, vol. 12 (The Clarendon Press Oxford University Press, New York, 1995)
K. Krohn, J. Rhodes, B. Tilson, Algebraic Theory of Machines, Languages, and Semigroups, ed. by M.A. Arbib. With a major contribution by K. Krohn, J.L. Rhodes, Chaps. 1, 5–9 (Academic Press, New York, 1968)
G. Lallement, Semigroups and Combinatorial Applications. Pure and Applied Mathematics, A Wiley-Interscience Publication (Wiley, New York, Chichester, Brisbane, 1979)
J. Rhodes, B. Steinberg, The q-theory of Finite Semigroups. Springer Monographs in Mathematics (Springer, New York, 2009)
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Steinberg, B. (2016). 1 The Structure Theory of Finite Monoids. In: Representation Theory of Finite Monoids. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-43932-7_1
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