  A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and
McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity
term, then the corresponding constraint satisfaction problem is tractable. A
binary operation is weak near-unanimity if and only if it is both commutative
and idempotent. Thus if the dichotomy conjecture is true, any finite
commutative, idempotent groupoid (CI groupoid) will be tractable. It is known
that every semilattice (i.e., an associative CI groupoid) is tractable. A
groupoid identity is of Bol-Moufang type if the same three variables appear on
either side, one of the variables is repeated, the remaining two variables
appear once, and the variables appear in the same order on either side (for
example, $x(x(yz))\approx(x(xy))z$). These identities can be thought of as
generalizations of associativity. We show that there are exactly 8 varieties of
CI groupoids defined by a single additional identity of Bol-Moufang type,
derive some of their important structural properties, and use that structure
theory to show that 7 of the varieties are tractable. We also characterize the
finite members of the variety of CI groupoids satisfying the self-distributive
law $x(yz)\approx(xy)(xz)$, and show that they are tractable.
