So comparing this to the previous sentence, does the (plural) definite article "na" make the difference between "cat na leabharlainne = the library's cat" vs. "cat leabharlainne = a library cat" (whatever a 'library cat' is?!) ?
That could explain why "leabhar leabharlainne = a library book" as opposed to "a library's book".
Of course it would then beg the question of why "hata fir = a man's hat" as opposed to "a man hat" (again, ignoring whatever that is for now!) ?
EDIT; I just thought of a 'man bag' which is a thing in English, so certainly such constructions can make sense in English at least some of the time.
Look above at scilling's reply to teeling2. If a noun is feminine, then in the genitive form you have "na" instead of "an". This info may be too basic for you but genitive is the form indicating belonging. So in English we add words, like we say "of the library", or we add an apostrophe s, "the library's", but in Irish they indicate it by changing the article to "na" (if the noun is feminine). I am very much a beginner here and I keep forgetting about this, and also having a hard time memorizing masculine vs. feminine nouns!
The genitive is indicated by using the genitive case of the noun - leabharlainne in thís case. If the noun is singular and feminine and definite, then the definite article is na, but only a small minority of genitive nouns are singular, feminine and definite.
The genitive form of a noun isn't always obvious - it can take some practice to get used to the patterns.
SatharnPHL I'm still working on grasping this; thanks for the info! Brigid, yes there is a lot I still don't understand about my own language. It is interesting though how learning another language can help you understand your own better. Learning Latin in college with all the noun and verb endings really helped me understand the different grammatic roles of words and how this can be done differently in different languages. Irish is so unique though so it's really a challenge.