  We present some conditions for social preference transitivity under the
majority rule when the individual preferences include cycles. First, our
concern is with the restriction on the preference orderings of individuals
except those (called cycle members) whose preferences constitute the cycles,
but the considered transitivity is, of course, of the society as a whole. In
our discussion, the individual preferences are assumed concerned and the cycle
members' preferences are assumed as strict orderings. Particularly, for an
alternative triple when one cycle is involved and the society is sufficient
large (at least 5 individuals in the society), we present a sufficient
condition for social transitivity; when two antagonistic cycles are involved
and the society has at least 9 individuals, necessary and sufficient conditions
are presented which are merely restricted on the preferences of those
individuals 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 axiomatic
manner. Connections between some already identified conditions and the
$\hat{O}\mbox{-}\hat{I}$ framework is examined.
