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In a statement issued at full-time, the Football Association said: "We are very concerned about the allegation of discrimination from an area of the crowd, which was reported to the match officials during the Premier League fixture between Liverpool and Bournemouth.
"Incidents of this nature have no place in our game, and we will work closely with the match officials, the clubs and the relevant authorities to establish the facts and ensure the appropriate action is taken."
Liverpool, external also issued a statement after the game: "Liverpool Football Club is aware of an allegation of racist abuse made during our Premier League game against AFC Bournemouth.
"We condemn racism and discrimination in all forms, it has no place in society or football.
"The club is unable to comment further as tonight's alleged incident is the subject of an ongoing police investigation, which we will support fully."
Reds head coach Arne Slot told Sky Sports: "The club made a clear statement.
"We should talk about this for a long time because we do not want this at Anfield.
It takes the shine off it (game) a bit because our fans were amazing, especially with the tributes to Diogo."
Premier League Match Centre, external during the match: "Tonight's match between Liverpool Football Club and AFC Bournemouth was temporarily paused during the first half after a report of discriminatory abuse from the crowd, directed at Bournemouth's Antoine Semenyo.
"This is in line with the Premier League's on-field anti-discrimination protocol.
We will continue to work with stakeholders and authorities to ensure our stadiums are an inclusive and welcoming environment for all."
Kick It Out said they "stand in solidarity" with Semenyo and added: "Thirty minutes into the first Premier League game of the season, and Bournemouth's Antoine Semenyo is racially abused by someone in the crowd.