Is a mandate valid to purchase an item when only the category (jins) is specified, such as 'a slave' or 'a garment'?

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The mandate is valid if one says, 'Buy for me a Turkish slave,' or 'a Harawi garment.' Similarly, it is valid if one says, 'Buy for me a slave,' or 'a garment,' without specifying the type (jins). The evidence supporting this is that if the agency to buy in totality is valid, it is valid for a part, analogous to the case of the slave. Furthermore, specifying the type implies permission to purchase the most expensive item of that type, thus reducing ambiguity, and stipulating the price can be detrimental, as the agent might not find an item at that price.