Supermatrix Representations of Semigroup Bands

Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rectangular band analogs and the wreath'' version of them are studied in detail. We introduce the fine’’ equivalence relations leading to ``multidimesional’’ eggbox diagrams. They are full images of Green’s relations on corresponding subsemigroups.