The left distributive law for a single binary operation is the law a(&) = (&)(a~). It has been studied in universal algebra (Stein [S], Kepka and Nemec [KN], Kepka [Kl, K2]; see also Jezek et al. [JKN] for a bibliography on the two-sided distributive law) and it has been studied more recently by set theorists because of its connection with elementary embeddings. For E, a limit ordinal let 6;. be the collection of all j: I’, -+ I’,, j an elementary embedding of (V,, E) into itself, j not the identity. Then the existence of a 3. such that c$;. # @ is a large cardinal axiom (see Gaifman [G] and Kanamori et ul. [KRS]). For Jo&;, let K~= cr(j), the critical point of j, and ti,+ i =j(~,~). Then 1 must equal sup(~,, : II <w) (Kunen [Ku]). Martin [Ml], Woodin [M2], Martin and Steel [MSl, MS2], and Woodin [W], in deep work, showed that slight strengthenings of this axiom imply determinacy properties of the real line, and subsequently that considerable weakenings suffice, making the axiom unnecessary for those determinacy properties. There is a natural operation . on 8;. (write UL: for u t’ in this and similar contexts below). For Jo gj., j extends to a map j: V,, , -+ I’,., , by defining, for A s V,, j(A) = U,,;,j(A n I’,). Then j may or may not be an elementary embedding of ( Vi + , , E) into itself, but at least j is elementary from (Vi, E, A) into (V,, E, jA). In the special case that A, as a set of ordered pairs, is a k E & we have that j(k) E &j-S Let j. k =j(k). Then the operation . on &j. is nonassociative, noncommutative, and left distributive. Another operation on 4. is composition: if k, IE&.. then k: IE&~.. Let Z be the set of laws a~(b~c)=(u~b)~~c, (ucb)c=u(bc), u(b’~c)=ub~.uc, a~ b = ub 0 a. Then &j~ satisfies Z, and Zimplies the left distributive law (u(bc)=(u~b)c=(uboa)c=ub(uc)). For Jo c!?~., let 4 be the closure of {j} under . . Let q, the set of “polynomials in j,” be the closure of (jj under . and 3. Results in which 4, 9j and their governing equations are involved appear in [Ml, M2, L], Dougherty