The precore: converse consistent enlargements and alternative axiomatic results

Since the precore violates (weak) converse consistency, two converse consistent enlargements are proposed. These two converse consistent enlargements are the smallest (weak) converse consistent solutions that contain the precore. On the other hand, we turn to a different notion of the reduction by considering the players and the activity levels simultaneously. Based on such revised reductions, we offer several axiomatizations of the precore.