I was reading something and I found this thing attached to a verb. The sentence was this one:
What meaning does it have?
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Closest English counterpart I can think of is, "what if" or "just imagine if"
ふと脇を見て、そこに洗濯物がたたんであったりしたら。 What if, I looked to my side, and all the landries were nicely folded right there. (how convenient would that be!)
It is a form of ommission, unique to Japanese. Imagine the full sentence being like:
ふと脇を見て、そこに洗濯物がたたんであったりしたら、すごくありがたい。 What if I looked to my side and all the laundries were nicely folded....that'd be handy!
What if I looked to my side and all the laundries were nicely folded....that'd be spooky! (assuming you are the only one in the house)
I have to see the context, so I can't give you the 100% translation - but I'd hazard a guess that - it's either someone lamenting how much laundry there is yet to be folded, or, cracking a joke about kind ghost that'd fold laundry for you.
したら is a conditional expression. This is a sentence (or a sentence fragment) that consists only of a long "if-clause". The corresponding main clause has been left out.
たり has two different roles:
On my days off, I read books or do something like this.
The correct translation would depend on the broader context, but this たり is probably used in the latter sense. It probably implies that 洗濯物がたたんである in this situation is a surprising (either positively or negatively) event to the speaker.
(I know this is unlikely but) In case [I/you/etc] happen to find the laundry folded in there...