For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in A*, thereby proving a conjecture due to Cavenagh and Drápal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith Normal Form of this matrix determines A*, so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g>0 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group.
We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks are bounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group.
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