Why is there t- ? Is there a linguistic name for that, is there an etymological reason behind it?
The reason it occurs now is when a masculine word with a vowel follows the singular definite article.
The linguistic name for that is “prothesis”. The etymological reason for it can be found here.
Please explain the link won't work
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why "an t-am" instead of just "an am" or "t-am"?
See kmradley’s thread above.
is, "na t-am" "the times" correct for the plural version ?
No — na hamanna is the plural of an t-am.