EU-citizens can transfer acquired rights from the Unemployment Insurance Scheme, when moving between EU/EEA-countries. This is stated in the EU-legislation, so it is a valid princip for all EU/EEA-countries.
However it is important that you become member of a "A-kasse" (Unemployment Insurance) in Slovakia at latest 8 weeks after you cease to be insured in a danish A-kasse. Maybe become member in Slovakia before you cancel membership in the danish A-kasse.
You need a PD U1 document issued by you danish A-kasse. This document can be shown to the Slovakia Unemployment Insurance, and here they can see for how long period you have paid to A-kasse in Denmark, and your registered working-periods. I will strongly recommend also to keep all your documentation (monthly payment notes) about your salary earned as a wage-earner in Denmark.
There can be additional requirements for being able to transfer rights to Slovakia Unemployment Insurance, but it is something you must ask the relevant Authorities about in Slovakia.
I can not tell if you can get Unemployment benefits from the Slovakia Insurance, in case you lose your job after 6 months in Slovakia. However you will be able to transfer your rights regarding membership-period and workperiods as mentioned - if there are further requirements in Slovakia, is something you must try to find out by asking the Unemployment Insurance in Slovakia.
Get up to 3 months unemployment benefits when go for jobsearch in another EU-country.
It is possible to take your unemployment benefits ("dagpenge") with you from the danish A-kasse for up to 3 months, BUT only if you go to another EU/EEA-country for looking for a job (jobsearch). You can read more about the requirements here: https://www.a-kasser.dk/transferring-unemployment-benefits.html.
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Anders Weber, www.a-kasser.dk