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Additional resources for An elementary theory of Eisenstein series
4 The bideterminant function The development of linear algebra over semirings certainly requires such an important matrix invariant as the determinant function, see . It turns out that the determinant cannot be deﬁned in a classical way even for matrices over commutative semirings without zero divisors. The main problem is connected with the fact that if a semiring is not a ring, then there are elements which do not possess additive inverses. 1 below) which has been known since 1972; see also  and [36, 38, 41, 57].
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Kim. Boolean Matrix Theory and Applications, volume 70 of Pure and Applied Mathematics. Marcel Dekker, New York, 1982.  V. N. Kolokol’tsov and V. Maslov. Idempotent Analysis and Applications. Kluwer, Dordrecht, 1997. Rank and determinant functions for matrices over semirings 33  J. Kuntzmann. Th´eorie des r´eseaux (graphes). Dunod, Paris, 1972. -K. Li and S. Pierce. Linear preserver problems. Amer. Math. Monthly, 108(7):591–605, 2001. -K. Li and N. K. Tsing. Linear preserver problems: a brief introduction and some special techniques.