$product[$x][$y] is short for $product[$x]->[$y]. In other words, $product[$x][$y] means $product[$x] needs to contain an array ref. However, you claim that $product[$x] contains the product name. $product[$x] cannot contain both an array ref and a product name, so you have a problem with the design of your data structure.

Perhaps you want something like:

$product[$x]{name}           is the name of product $x.
$product[$x]{opts}[$i]       is the option $i of product $x.

@product                     is the list of products.
map { $_->{name} } @product  is the list of product names.
@{$product[$x]{opts}}        is the list of options of product $x.
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or maybe you'd like to use a hash of products instead of an array of products:

$product{$name}[$i]   is the option $i of product $name.

keys %product         is the list of product names.
@{$product{$name}}    is the list of options of product $name.
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(Product names must be unique when using a hash of products.)