Titlebook: Representation Theory of Finite Monoids; Benjamin Steinberg Textbook 2016 Springer International Publishing Switzerland 2016 automata theo

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11 The Rhodes Radical and Triangularizabilitye are precisely the monoids with a basic algebra over . and so it also characterizes these monoids. Our treatment of these topics is based, for the most part, on that of Almeida, Margolis, the author, and Volkov?[AMSV09].

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Aromatic

2 ,-trivial Monoidsoids, which have quite recently found applications in the analysis of Markov chains; see Chapter?. The first section of this chapter discusses lattices and prime ideals of arbitrary finite monoids, as they shall play an important role in both the general theory and in the structure theory of .-trivi

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3 Inverse Monoidsst important class of monoids outside of groups. They abstract the notion of partial symmetry in much the same way that groups abstract the notion of symmetry. For a detailed discussion of this viewpoint, see Lawson [Law98]. From the perspective of this book they provide a natural class of monoids w

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4 Recollement: The Theory of an Idempotent the algebras . and .∕., for an idempotent .?∈?., known as .?[., ., .]. We first learned of this subject from the monograph of Green?[., Chapter?.]. A presentation much closer in spirit to ours is that of Kuhn?[.]. In the next chapter, we shall apply this theory to construct the irreducible represen

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5 Irreducible Representationsbetween equivalence classes of irreducible representations of a finite monoid and equivalence classes of irreducible representations of its maximal subgroups (taken one per regular .-class). We follow here the approach of?[.], using the techniques of Chapter?. Let us commence by introducing formally

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6 The Grothendieck Ringhic to the direct product of the Grothendieck rings of its maximal subgroups (one per regular .-class). This result was first proved by McAlister for . in the language of virtual characters?[.]. In Chapter?., the results of this chapter will be used to study the ring of characters and the character