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

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