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Coxeter groups possess an associative operation, called variously theDemazure, greedy, or $0$-Hecke product. For symmetric groups, this product hasan amusing formulation as matrix multiplication in the min-plus (tropical)semiring of two matrices associated to the permutations. We prove that thismin-plus formulation extends to furnish a Demazure product on a much largergroup of integer permutations, consisting of all permutations that change thesign of finite many integers. We prove several alternative descriptions of thisproduct and some useful properties of it. These results were developed inservice of future applications to Brill-Noether theory of algebraic andtropical curves; the connection is surveyed in an appendix.

An extended Demazure product on integer permutations via min-plus matrix multiplication