  One-parameter semigroups of antitriangle idempotent supermatrices and
corresponding superoperator semigroups are introduced and investigated. It is
shown that $t$-linear idempotent superoperators and exponential superoperators
are mutually dual in some sense, and the first gives additional to exponential
solution to the initial Cauchy problem. The corresponding functional equation
and analog of resolvent are found for them. Differential and functional
equations for idempotent (super)operators are derived for their general $t$
power-type dependence.
