Supertropical \(\mathrm{SL}_n\). (English) Zbl 1386.15008
Summary: Extending earlier work on supertropical adjoints and applying symmetrization, we provide a symmetric supertropical version \(\mathrm{SLS}_n\) of the special linear group \(\mathrm{SL}_n\), which we partially decompose into submonoids, based on ‘quasi-identity’ matrices, and we display maximal sub-semigroups of \(\mathrm{SLS}_n\). We also study the monoid generated by \(\mathrm{SLS}_n\) and its natural submonoids. Several illustrative examples are given of unexpected behavior. We describe the action of elementary matrices on \(\mathrm{SLS}_n\), which enables one to connect different matrices in \(\mathrm{SLS}_n\), but in a weaker sense than the classical situation.
MSC:
| 15A04 | Linear transformations, semilinear transformations |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 16Y60 | Semirings |
| 14T05 | Tropical geometry (MSC2010) |
| 22E15 | General properties and structure of real Lie groups |
Keywords:
supertropical matrix algebra; special linear monoid; matrix monoid; tropical adjoint matrix; quasi-identity; elementary matrixReferences:
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