Supertropical $\operatorname{SL}_n$

Extending earlier work on supertropical adjoints and applying symmetrization, we provide a symmetric supertropical version $\operatorname {SLS}_n$ of the special linear group, which we partition into submonoids, based on "quasi-identity" matrices, and we display maximal sub-semigroups of $\operatorname {SLS}_n$. We also study the monoid generated by $\operatorname {SLS}_n$. Several illustrative examples are given of unexpected behavior. We describe the action of elementary matrices on $\operatorname {SLS}_n$, which enables one to connect different matrices in $\operatorname {SLS}_n$, but in a weaker sense than the classical situation.