# Return an induced sub-poset
	#
	# The elements of the result are those given in argument.
	#
	# ~~~
	# var pos = new POSet[String]
	# pos.add_chain(["A", "B", "C", "D", "E"])
	# pos.add_chain(["A", "X", "C", "Y", "E"])
	#
	# var pos2 = pos.sub(["A", "B", "D", "Y", "E"])
	# assert pos2.has_exactly(["A", "B", "D", "Y", "E"])
	# ~~~
	#
	# The full relationship is preserved between the provided elements.
	#
	# ~~~
	# for e1 in pos2 do for e2 in pos2 do
	#    assert pos2.has_edge(e1, e2) == pos.has_edge(e1, e2)
	# end
	# ~~~
	#
	# Not that by definition, the direct relationship is the transitive
	# reduction of the full reduction. Thus, the direct relationship of the
	# sub-poset may not be included in the direct relationship of self because an
	# indirect edge becomes a direct one if all the intermediates elements
	# are absent in the sub-poset.
	#
	# ~~~
	# assert pos.has_direct_edge("B", "D")  == false
	# assert pos2.has_direct_edge("B", "D") == true
	#
	# assert pos2["B"].direct_greaters.has_exactly(["D", "Y"])
	# ~~~
	#
	# If the `elements` contains all then the result is a clone of self.
	#
	# ~~~
	# var pos3 = pos.sub(pos)
	# assert pos3 == pos
	# assert pos3 == pos.clone
	# ~~~
	fun sub(elements: Collection[E]): POSet[E]
	do
		var res = new POSet[E]
		for e in self do
			if not elements.has(e) then continue
			res.add_node(e)
		end
		for e in res do
			for f in self[e].greaters do
				if not elements.has(f) then continue
				res.add_edge(e, f)
			end
		end
		return res
	end