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An associative operation $\\star$ may be defined on $\\mathcal M_{mn}$ by $X\\star Y=XAY$ for all $X,Y\\in\\mathcal M_{mn}$, and the resulting \\emph{sandwich semigroup} is denoted $\\mathcal M_{mn}^A=\\mathcal M_{mn}^A(\\mathbb F)$. These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. 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