ST. PAUL — Gov. Mark Dayton said he and his next lieutenant governor had a cordial lunch as they talked over how they will work together despite party differences.

State Senate President Michelle Fischbach, a Republican, is becoming Dayton’s lieutenant by law as a result of him appointing Tina Smith to temporarily replace Al Franken in the U.S. Senate.

Dayton, a Democrat, said he and Fischbach talked briefly at Friday’s lunch about her desire to keep her Senate seat. Fischbach and state Republicans have said they believe state law allows her to hold both jobs.

Democrats disagree and have asked for an attorney general’s opinion.

The matter may wind up in court because Republicans could lose their Senate majority of Fischbach has to give up the seat.