The first omitting cardinal for Magidority

An infinite cardinal λ is Magidor if and only if . It is known that if λ is Magidor then for some , and the first such α is denoted by . In this paper we try to understand some of the properties of . We prove that can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of being a successor of a singular cardinal with uncountable cofinality.