Could you recommend an open source library written in F# which provides generic types for FA construction and basic algorithms (NFA to DFA transformation, FA minimization ...)?
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My first thought would be to use the state machine monad, which has be blogged/written/discussed/produced several times, a couple of references are below.
http://fsharpcode.blogspot.com/2008/12/f-state-monad-type-state-state-state-of.html
http://codebetter.com/blogs/matthew.podwysocki/archive/2009/12/30/much-ado-about-monads-state-edition.aspx
although this doesn't give you the translations you want... Might be a place to start.

Snark
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It does not fulfil all your requirements: It's in C#, and it does not seem to support translation to DFA or minimization. At least it's usable from F# and could be used as a starting point. I've never used it myself, so I can't vouch for its suitability or quality.
See also NFA/DFA implementation in C#