nLab quasiregular rig

This article is about the $*$-rig that is the generalization of a Kleene star algebra. For the $*$-rig that is an $\mathbb{N}$-algebra with an anti-involution, see star-algebra.

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 Definition

A rig $R$ is a quasiregular rig or quasiregular semiring or closed rig or closed semiring or Lehmann rig or Lehmann semiring or $*$-rig or $*$-semiring if every element in $R$ is a quasiregular element; equivalently, $R$ is quasiregular if there is a function $(-)^*:R \to R$ such that for all elements $r \in R$, $r^* = r^* r + 1$ and $r^* = r r^* + 1$.

 Examples

• Assuming excluded middle, the set of non-negative extended reals $[0, \infty]$ together with the usual addition and multiplication of reals is a quasiregular rig with the star operation given by $a^* = \frac{1}{1 - a}$ for $0 \leq a \lt 1$ and $a^* = \infty$ for $a \geq 1$.

• The only quasiregular ring is the trivial ring, because if the multiplicative neutral element $1$ is quasiregular, then we have $1^* = 1^* 1 + 1$ and $1^* = 1^* + 1$ and thus $0 = 1$.

• More generally, the only quasiregular rig with cancellative addition is the trivial rig, because if the multiplicative neutral element $1$ is quasiregular, then we have $1^* = 1^* 1 + 1$ and $1^* = 1^* + 1$ and thus $0 = 1$ by the cancellative property of addition.

• A commutative rig is quasiregular if and only if it satisfies the Conway product-star axiom: for all $r \in R$ and $s \in R$, $(r s)^* = 1 + r (s r)^* s$.

Related concepts

• rig, semiring

• Conway semiring

• Kleene star algebra

• quasiregular element

References

• Wikipedia, Quasiregular element#Generalization to semirings

• Wikipedia, Semiring#Star semirings

• Daniel J. Lehmann, Algebraic structures for transitive closure, Theoretical Computer Science, volume 4, issue 1, pages 59–76, 1977 (doi:10.1016/0304-3975(77)90056-1, pdf)

• Dexter Kozen (1990). On Kleene algebras and closed semirings. In: Branislav Rovan (editor). Mathematical Foundations of Computer Science 1990. MFCS 1990. Lecture Notes in Computer Science, vol 452. Springer, Berlin, Heidelberg. (doi:10.1007/BFb0029594, pdf)

• Manfred Droste, Werner Kuich, Semirings and Formal Power Series (2009). In: Manfred Droste, Werner Kuich, Heiko Vogler, (eds) Handbook of Weighted Automata. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. pp. 3–28. (doi:10.1007/978-3-642-01492-5_1)