The Supreme Court on Monday turned away Republican challenges to the presidential election results in Pennsylvania, refusing to take up a months-long dispute over extending the deadline in that state for receiving mail-in ballots.

It was part of a purge of sorts. The high court formally dismissed a range of suits filed by Donald Trump and his allies in Pennsylvania, Wisconsin, Michigan, Georgia and Arizona — all states won by Democrat Joe Biden. The court’s intent in most of those had been signaled when it refused to expedite consideration of them before Biden was inaugurated as president.

The case about deadlines for receiving mail-in ballots was different, though. Three justices — Clarence Thomas, Samuel A. Alito Jr. and Neil M. Gorsuch — said it deserved the court’s attention, even though the number of votes at issue would not call into question Biden’s victory.

“A decision in these cases would not have any implications regarding the 2020 election,” Alito wrote. “But a decision would provide invaluable guidance for future elections.”

It takes the votes of four justices to accept a case for review. Although changing election rules because of the pandemic has been a theme of Republican challenges in the wake of Trump’s defeat, the rest of the conservative majority was silent.

Neither Chief Justice John G. Roberts Jr. nor two of the three justices nominated by Trump signed on to dissents from Thomas and Alito. Besides Gorsuch, Trump chose Justices Brett M. Kavanaugh and Amy Coney Barrett.

The issue is whether state courts or other officials have the right to change voting procedures set by the legislature where federal elections are at stake. In extending the right to a mail-in ballot to all voters, Pennsylvania’s Republican-controlled legislature said the ballots must be received by 8 p.m. on Election Day to be counted.

But the state’s Democrats challenged that. Citing the pandemic and concerns about the Postal Service’s ability to deliver mail on time, the Pennsylvania Supreme Court extended the receipt deadline until three days after the election. It cited a provision in the state constitution promising fair elections.

In a pre-election challenge, the Supreme Court was deadlocked, meaning the extension applied. In the end, it affected fewer than 10,000 votes, and Biden won by about 80,000.

But the question of who decides voting procedures has become an important one for Republicans, who control more of the state’s legislatures.

Thomas, Alito, Gorsuch and Kavanaugh all endorsed a view that the Constitution’s command that the “legislature” design the rules of elections means that state courts and agencies do not have a free hand in making changes to state laws. They say federal courts have a role in overseeing the state court decisions.

That is a position that received only three votes when the Supreme Court in 2000 decided Bush v. Gore. But, as Gorsuch has written, “The Constitution provides that state legislatures — not federal judges, not state judges, not state governors, not other state officials — bear primary responsibility for setting election rules.”

Kavanaugh endorsed that view, as well, in the pre-election challenge. But he did not join the others Monday who wanted to take the Pennsylvania case, which the court’s majority said was moot.

Roberts has set himself apart from the other conservatives, saying state courts have a role to play. And new Justice Amy Coney Barrett has so far not taken a position publicly.

Thomas said it was a mistake to ignore the controversy.

“These cases provide us with an ideal opportunity to address just what authority nonlegislative officials have to set election rules, and to do so well before the next election cycle,” he wrote. “The refusal to do so is inexplicable.”

Richard H. Pildes, an election law expert at New York University, said the decision likely was motivated by the court’s desire to provide closure to the election.

“I view the court as deciding to refrain from taking any action that could be misconstrued — and surely would be, by some — as casting any doubt on the legitimacy of the 2020 election,” Pildes wrote in an email.

“But the issues are major ones the court is inevitably going to have to confront in upcoming federal elections; without clear resolution, federal elections will continually be roiled by these issues.”

Indeed, Republican officials have cited the concern about election rules as reason not to declare the recent election fair.

“There were a few states that did not follow their state laws, that’s really the dispute that you’ve seen continue on,” House Minority Whip Steve Scalise (R-La.) said Sunday on ABC’s “This Week.”

The Supreme Court has uniformly rejected challenges to the election results.

On Dec. 11, it dismissed a bid by Trump and the state of Texas to overturn the results in those four battleground states won by Biden, blocking the president’s legal path to reverse his reelection loss.

The court’s unsigned order was short, and it denied Texas’s request to sue the states over how they conducted their elections. Texas has not shown it has a legal interest “in the manner in which another state conducts its elections,” the order said. It dismissed all pending motions about the case.

While Roberts has drawn the ire of the right for the rejection, no justice on the court said they would have granted Texas the remedy it sought, which was to disallow the electors certified by the states.

Alito and Thomas said they did not think the court had the authority to simply reject a state’s filing and not consider the issue, but both said they were not prepared to grant the relief that Texas sought.

Earlier that week, the court without noted dissent had rejected a similar request from Republicans in Pennsylvania. Conservatives hold a ­6-to-3 majority on the court, and Trump has nominated three of the justices.

After the decisions, Trump said the Supreme Court “really let us down” and has expanded his criticism of the justices since then.