Algebraically closed abelian $l$-groups

Rok, strany: 2015, 841 - 862

Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic $1$. Semifields of characteristic $1$ are equivalent to abelian lattice-ordered groups. It is proved that such a semifield $A$ is algebraically closed if and only if the pure equations $xⁿ=a$ and certain quadratic equations are solvable in $A$. Using a sheaf representation for $z$-projectable abelian $l$-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic $1$ is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic $1$. While polynomials satisfy the ``fundamental theorem of algebra'', the multiplicative structure of rational functions is determined by means of ``divisors'', extracted from the additive structure of $A$ modulo parallelogram identities.

ISO 690:

Rump, W. 2015. Algebraically closed abelian $l$-groups. In Mathematica Slovaca, vol. 65, no.4, pp. 841-862. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0059

APA:

Rump, W. (2015). Algebraically closed abelian $l$-groups. Mathematica Slovaca, 65(4), 841-862. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0059