Sounds like you are talking about slips. There are also suckers that grow up from the roots.
Here's more.
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There are four kinds of propagation material on pineapple plants: ratoon suckers arise below ground, suckers originate in the leaf axils, slips grow from the fruit itself or along the stalk below the fruit and crowns are the leafy tops of the fruit. All four types work, although slips and suckers are preferred in commerce. Fortunately, each pineapple fruit in the supermarket comes with a crown which can be used to start the plant which will develop slips and suckers for subsequent use.
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