Every toric map $f : X \to Y$ corresponds to a unique map $g : N_X \to N_Y$ of lattices such that, for every cone $\sigma$ in the fan of $X$, there is a cone in the fan of $Y$ that contains the image $g(\sigma)$. For more information on this correspondence, see Theorem 3.3.4 in Cox-Little-Schenck's Toric Varieties. This method returns an integer matrix representing $g$.
We illustrate how to access this defining feature of a toric map with the projection from the second Hirzebruch surface to the projective line.
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An inclusion map into the Cartesian square of a normal toric variety corresponds to matrix having identity and zero blocks.
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In a well-defined toric map, the number of rows in the underlying matrix must equal the dimension of the target and the number of columns must equal the dimension of the source.
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The output display for toric maps is inherited the underlying map of lattices.
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Since this is a defining attribute of a toric map, no computation is required.