CHICAGO — Mark Buehrle is staying with the White Sox, agreeing yesterday to a $56 million, four-year contract extension through the 2011 season. Fans at yesterday’s game were told of the deal over the public-address system as the White Sox left the field following a 6–3 win over the Minnesota Twins. In the eighth inning, Buehrle was seen in the dugout hugging manager Ozzie Guillen and his teammates. Buehrle, a three-time All-Star, was eligible to become a free agent after this season. With the White Sox struggling this season just two years after winning the World Series, Buehrle’s name popped up often in trade rumors. He is 6–4 this season with a 3.03 ERA. He has a $9.5 million salary this year and will earn $14 million annually during the next four seasons. Talks had gone on for weeks, and the sides had been said to be haggling over a no-trade provision. “There is no question what Mark Buehrle means to this organization, our clubhouse and our fans,” the White Sox general manager, Ken Williams, said. “He is one of the very best left-handers in the game today. We cannot be happier to reach this agreement with Mark, and now look forward to this team winning games the remainder of this season and in future years with Mark on the mound.”