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Constructions Involving Involutary Semirings and Rings

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Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2188))

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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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Notes

  1. 1.

    Such a structure is called a K-ring in Cohn [31, Sect. 1].

  2. 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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  1. Département de Mathématiques, Université de Caen Normandie, Caen, France

    Friedrich Wehrung

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Wehrung, F. (2017). Constructions Involving Involutary Semirings and Rings. In: Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups. Lecture Notes in Mathematics, vol 2188. Springer, Cham. https://doi.org/10.1007/978-3-319-61599-8_6

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