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We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n × n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2×2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of "chain structured tropical matrix semigroups"; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3 × 3 upper triangular matrix semigroups, and a new proof of the fact that this semigroup satisfies the same identities as the bicyclic monoid.

Previous studies have documented a gender gap in the study of economics in Canada, the UK, and the US. One important factor may be women's low expectations about their ability to succeed in economics courses. Women in our sample expect to do less well than men in an introductory microeconomics course, even after controlling for variables relating to family background, academic experience, and mathematics experience. These expectations are partly self-fulfilling, since expected grades have an important and positive effect on class performance. We also find that having taken an economics course in secondary school actually has a negative effect on performance. We observe this negative effect for women and men, but it is more pronounced for women. When we control for both expectations and secondary-school experience with economics, the independent effect of gender is small and insignificant.