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  "body": "\n## Summary\n\nA partially well-ordered set is a [[partially ordered set|0QV2JS9RJ3]] whose order is a [[well-quasi-ordering|NH25G6GC07]].\nSuch an order is also called a well-partial-order.\nIt generalizes a [[well-order|01CGSJ0GNQ]]: the order need not be total, but it is well-founded and every antichain is finite.\nOne also says that a subset of a preordered set is partially well-ordered when the restricted order has this property.\n\n## Explanations\n\n### L1 human\n\nA well-order lines elements up so that every descent stops.\nA partial well-order allows elements that cannot be compared, but only finitely many at a time in any antichain, and still no endless descent.\nThe set $\\N \\times \\N$ ordered coordinatewise is an example: $(1, 2)$ and $(2, 1)$ are incomparable, yet every antichain is finite.\n",
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