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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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