The offspring of a parent cultivar will likely be just the offspring of a parent cultivar, not a strain. A strain is a group of similar lilies with similar genetics, so similar that when crossed they still produce the same similarities. Most cultivars are not selections within strains; they are chosen from the progeny of crosses of two dissimilar parents. Therefore the great variation in genetics from a cultivar predicts dissimilar offspring, which by definition is not a strain. And let's not forget the self infertility factor: with what lily did said cultivar cross?
But the gist of what Lorn says is correct, that one can save seed from a virus infected lily, and produce virus free babies. And that these babies can no longer be called solely by the parent cultivar's name because they are not the same.
So, for example, if you saved seed from a virused Ariadne lily (Lilium 'Ariadne'), the seedlings produced can no longer be called Ariadne lilies (or Lilium 'Ariadne').
They could be designated (as examples):
seedlings from Ariadne lily, or
seedlings of Ariadne lily, or
Ariadne lily seedlings, or
Ariadne lily offspring, or
Lilium ex 'Ariadne' - (not to be confused with "x" or "×" (meaning: cross), "ex" is a Latin word (meaning: out of, from).
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a little more trivia:
"×" or "x" (as in Lilium 'Lollipop' × Lilium 'Cathedral Windows') is technically pronounced "cross".
"ex" (as in Lilium ex 'Ariadne') is pronounced "eks"-(like the letter X). (It's as close as we come to Latin pronunciation in the English language.)
