self-distributive operation

Self-distributive operations


Let :M×MM\cdot\colon M \times M\to M be a binary operation, i.e. (M,)(M,\cdot) is a magma. We say that the operation \cdot is

  • left self-distributive if for all x,y,zMx,y,z\in M, x(yz)=(xy)(xz)x\cdot(y\cdot z) = (x\cdot y)\cdot (x\cdot z);
  • right self-distributive if for all x,y,zMx,y,z\in M, (yz)x=(yx)(zx)(y\cdot z)\cdot x = (y\cdot x)\cdot (z\cdot x).


  • The binary operation in any semilattice is self-distributive on both sides, following from associativity, commutativity, and idempotence.

  • The operations in a rack (and hence also in a quandle) are self-distributive on the side on which they act. In particular, this includes the operation of conjugation in a group.

  • A Laver table? is the multiplication table of a self-distributive operation.

Revised on September 9, 2011 01:12:59 by Mike Shulman (