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The way I learnt it was that traditionally the forms with Dé refer to specific instances of the day, i.e., Dé Luain seo chugainn "this coming Monday", Dé Domhnaigh seo caite "last Sunday".
Without seo chugainn or seo caite, you have to work it out from context of the verb, etc. Dé Máirt means "on Tuesday", but a specific one.
The form with the article is used for the day more generally so that ar an Luan is more like "on Mondays" or "every Monday" than a particular date.
The exception to that usage is Déardaoin, from which Dé can't be separated, but you just add the article to it for the more general meaning, i.e, ar an Déardaoin.
That was how I was taught it but the usage may be changing ...
_________________ Múinteoir Gaeilge - Irish Teacher My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation. Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí. Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).
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