A note on decomposition of idempotent uninorms into an ordinal sum of singleton semigroups

The idempotent uninorms are characterized by means of the ordinal sum of Clifford. It is shown that idempotent uninorms are in one-to-one correspondence with special linear orders on [0,1][0,1]. A connection between respective linear order on [0,1][0,1] and the characterizing multi-function of the uninorm is also investigated.