  In some varieties of algebras one can reduce the question of finding most
general unifiers (mgus) to the problem of the existence of unifiers that
fulfill the additional condition called projectivity. In this paper we study
this problem for Fregean (1-regular and orderable) varieties that arise from
the algebraization of fragments of intuitionistic or intermediate logics. We
investigate properties of Fregean varieties, guaranteeing either for a given
unifiable term or for all unifiable terms, that projective unifiers exist. We
indicate the identities which fully characterize congruence permutable Fregean
varieties having projective unifiers. In particular, we show that for such a
variety there exists the largest subvariety that have projective unifiers.
