  In an earlier work, the author observed that Boolean inverse semi-groups,
with semigroup homomorphisms preserving finite orthogonal joins, form a
congruence-permutable variety of algebras, called biases. We give a full
description of varieties of biases in terms of varieties of groups: (1) Every
free bias is residually finite. In particular, the word problem for free biases
is decidable. (2) Every proper variety of biases contains a largest finite
symmetric inverse semigroup, and it is generated by its members that are
generalized rook matrices over groups with zero. (3) There is an
order-preserving, one-to-one correspondence between proper varieties of biases
and certain finite sequences of varieties of groups, descending in a strong
sense defined in terms of wreath products by finite symmetric groups.
