Hi,
yes, as an EU-citizen it is possible to transfer working periods/income and member/insurance periods from Ireland to the danish A-kasse.
However in Denmark we have a special condition for being able to transfer rights:
You must within 8 weeks after you ceased to be insured in Ireland, start a work here for at least 296 hours over 12 weeks / 3 months.
I don't know how payments to unemployment insurance work in Ireland. If you have stopped paying to the irish insurance, then you must start a job in Denmark within 8 weeks after you stopped payments, for being able to transfer your earned rights from the irish unemployment insurance.
According to EU law you can only be member of one A-kasse/unemployment Insurance at a time, and that must be in the EU-country where you work.
For transfering rights you need a PD U1 document from the Irish Unemployment Insurance. This docuement tells the danish A-kasse about your earned rights.
Read more here: https://www.a-kasser.dk/transfering-periods-of-work-and-insurance-from-another-eea-country/
Best regards,
Anders Weber, www.a-kasser.dk