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We construct a constant size population model allowing for general selectiveinteractions and extreme reproductive events. It generalizes the idea of (Kroneand Neuhauser 1997) who represented the selection by allowing individuals tosample potential parents in the previous generation before choosing the'strongest' one, by allowing individuals to use any rule to choose their realparent. Via a large population limit, we obtain a generalisation of$\Lambda$-Fleming Viot processes allowing for non transitive interactionsbetween types. We provide fixation properties, and give conditions for theseprocesses to be realised as solutions of stochastic differential equations.

Multidimensional $\Lambda$-Wright-Fisher processes with general frequency-dependent selection