Collected Examples of Monoids, (Pre)-Semirings and Dioids

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The present chapter is intended as a catalogue of examples covering the various algebraic structures studied throughout this book.

The successive items to be found in this chapter are: in Sect. 1, examples of monoids with emphasis on canonically ordered monoids; in Sect. 2, examples of presemirings and pre-dioids which are not semirings; in Sect. 3, examples of semirings (and rings) which are not dioids; and in Sect. 4, examples of dioids.

Using the many examples stated in this chapter, a virtually unlimited number of other examples may, of course, be derived by homomorphism, but also by considering matrices, polynomials, formal series on these algebraic structures.Verification of the main properties enjoyed by the various examples stated here is either explicitly stated in the text of the chapter, or can be found in the previous chapters.

Throughout this chapter, for most of the examples considered, we provide references to the chapters (in bold character) and sections or subsections dealing with each particular example.