Wisconsin will hold a special election on January 16, 2018, to fill the vacant State Senate seat, 10th district. Libertarian Brian Corriea is the only candidate who filed for the Libertarian primary, so he is deemed nominated. The identity of the Republican and Democratic nominees will be determined in a special primary for each party being held in December. See this story.
In Wisconsin, all candidates for partisan nomination appear on the same ballot, and a voter chooses a party on the ballot. Write-in spaces are also available, but the barrier to qualifying by write-in when there is no on-ballot candidate is the number of signatures required to qualify (400 in this case).
IF I understand correctly, there will be the opportunity on the primary ballot to select among the Republican, Democratic, Wisconsin Green (WIG), Libertarian, and Constitution parties. To be nominated, a WIG or Constitution write-in would need 400 votes, while the Libertarians, Republicans, or Democrats could nominate a write-in candidate who finishes first, regardless of the votes received.
The prospective Socialist candidate in AD-58 (West Bend) was running as an independent. Wisconsin permits to independents to provide a party name or slogan. If he had qualified (it does not appear that he submitted a petition), he would have been placed on the special election ballot, skipping the special primary.
Corriea will be on the primary ballot, as the only candidate listed for the Libertarian nomination. Technically he could lose the nomination to a write-in candidate, but of course that’s not going to happen and there are no other Libertarians running. Wisconsin doesn’t cancel the primary when there’s only one candidate though, there will still be the Libertarian space on the primary ballot with Corriea’s name and a write-in space.