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2017 | OriginalPaper | Chapter

6. Constructions Involving Involutary Semirings and Rings

Author : Friedrich Wehrung

Published in: Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Publisher: Springer International Publishing

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Abstract

The axioms of ring theory, when deprived of the existence of additive inverses, yield the axioms of semirings. When endowed with an additional involutary anti-automorphism (we will talk about involutary semirings), semirings will enjoy quite a fruitful interaction with Boolean inverse semigroups, the basic idea being to have the multiplications agree and the inversion map correspond to the involution.

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previous chapter Type Theory of Special Classes of Boolean Inverse Semigroups
next chapter Discussion
Footnotes
1
Such a structure is called a K-ring in Cohn [31, Sect. 1].
 
2
Due to a conflict of notation and intuitive meaning with the notations \(\mathop{\mathrm{\mathbf{d}}}\nolimits\) and \(\mathop{\mathrm{\mathbf{r}}}\nolimits\), for the domain and the range, in inverse semigroups, we kept “\(\mathop{\mathrm{\mathbf{s}}}\nolimits\)” for the source but we changed the usual “\(\mathop{\mathrm{\mathbf{r}}}\nolimits\)” to “\(\mathop{\mathrm{\mathbf{t}}}\nolimits\)” for the target.
 
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Metadata
Title
Constructions Involving Involutary Semirings and Rings
Author
Friedrich Wehrung
Copyright Year
2017
Book
Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Print ISBN: 978-3-319-61598-1

Electronic ISBN: 978-3-319-61599-8

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-61599-8_6

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