## The left distributive law and the freeness of an algebra of elementary embeddings.(English)Zbl 0822.03030

The left distributive law for a single binary operation is the law $$a(bc)= (ab) (ac)$$. It has been studied in universal algebra and it has been studied more recently by set theorists because of its connection with elementary embeddings. For $$\lambda$$ a limit ordinal let $${\mathcal E}_ \lambda$$ be the collection of all $$j: V_ \lambda\to V_ \lambda$$, $$j$$ an elementary embedding of $$(V_ \lambda, \in)$$ into itself, $$j$$ not the identity. Then the existence of a $$\lambda$$ such that $${\mathcal E}_ \lambda \neq\emptyset$$ is a large cardinal axiom. For $$j\in {\mathcal E}_ \lambda$$ let $$\kappa_ 0= \text{cr} (j)$$, the critical point of $$j$$, and $$\kappa_{n+1}= j(\kappa_ n)$$. Then $$\lambda$$ must equal $$\sup\{ \kappa_ n$$: $$n<\omega\}$$.
There is a natural operation $$\cdot$$ on $${\mathcal E}_ \lambda$$ (write $$uv$$ for $$u\cdot v$$ in this and similar contexts below). For $$j\in {\mathcal E}_ \lambda$$, $$j$$ extends to a map $$j: V_{\lambda+1}\to V_{\lambda+1}$$ by defining, for $$A\subseteq V_ \lambda$$, $$j(A)= \bigcup_{\alpha< \lambda} j(A\cap V_ \alpha)$$. Then $$j$$ may or may not be an elementary embedding of $$(V_{\lambda+1}, \in)$$ into itself, but at least $$j$$ is elementary from $$(V_ \lambda, \in, A)$$ into $$(V_ \lambda, \in, jA)$$. In the special case that $$A$$, as a set of ordered pairs, is a $$k\in {\mathcal E}_ \lambda$$, we have that $$j(k)\in {\mathcal E}_ \lambda$$. Let $$j\cdot k= j(k)$$. Then the operation $$\cdot$$ on $${\mathcal E}_ \lambda$$ is nonassociative, noncommutative, and left distributive.
Another operation on $${\mathcal E}_ \lambda$$ is composition: if $$k,l\in {\mathcal E}_ \lambda$$, then $$k\circ l\in {\mathcal E}_ \lambda$$. Let $$\Sigma$$ be the set of laws $$a\circ (b\circ c)= (a\circ b)\circ c$$, $$(a\circ b)c= a(bc)$$, $$a(b\circ c)= ab\circ ac$$, $$a\circ b= ab\circ a$$. Then $${\mathcal E}_ \lambda$$ satisfies $$\Sigma$$, and $$\Sigma$$ implies the left distributive law $$(a(bc)= (a\circ b)c= (ab\circ a)c= ab (ac))$$.
For $$j\in {\mathcal E}_ \lambda$$, let $${\mathcal A}_ j$$ be the closure of $$\{j\}$$ under $$\cdot$$. Let $${\mathcal P}_ j$$, the set of “polynomials in $$j$$”, be the closure of $$\{j\}$$ under $$\cdot$$ and $$\circ$$. A natural question, noticed independently by a number of people, is whether $${\mathcal A}_ j$$ and $${\mathcal P}_ j$$ are the one-generated free algebras $${\mathcal A}$$ and $${\mathcal P}$$ subject to, respectively, the left distributive law and to $$\Sigma$$. In this paper a normal form theorem for the free algebras is proved, from which the freeness of $${\mathcal A}_ j$$ and $${\mathcal P}_ j$$ can be derived.

### MSC:

 03E55 Large cardinals 08B20 Free algebras
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### References:

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