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We present some conditions for social preference transitivity under themajority rule when the individual preferences include cycles. First, ourconcern is with the restriction on the preference orderings of individualsexcept those (called cycle members) whose preferences constitute the cycles,but the considered transitivity is, of course, of the society as a whole. Inour discussion, the individual preferences are assumed concerned and the cyclemembers' preferences are assumed as strict orderings. Particularly, for analternative triple when one cycle is involved and the society is sufficientlarge (at least 5 individuals in the society), we present a sufficientcondition for social transitivity; when two antagonistic cycles are involvedand the society has at least 9 individuals, necessary and sufficient conditionsare presented which are merely restricted on the preferences of thoseindividuals except the cycle members. Based on the work due to Slutsky (1977)and Gaertner \& Heinecke (1978), we then outline a conceptual$\hat{O}\mbox{-}\hat{I}$ framework of social transitivity in an axiomaticmanner. Connections between some already identified conditions and the$\hat{O}\mbox{-}\hat{I}$ framework is examined.

Conditions for Social Preference Transitivity When Cycle Involved and A $\hat{O}\mbox{-}\hat{I}$ Framework