Axiom Of Choice

We have the following indirect implication of form equivalence classes:

85 \(\Rightarrow\) 268 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
85 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 61	clear
61 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 268	Subalgebra lattices of unary algebras and an axiom of choice, Lampe, W. A. 1974, Colloq. Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
85:	\(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
61:	\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function.
88:	\(C(\infty ,2)\): Every family of pairs has a choice function.
268:	If \({\cal L}\) is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and \(\alpha \) is an automorphism of \({\cal L}\) of order 2 (that is, \(\alpha ^{2}\) is the identity) then there is a unary algebra \(\frak A\) and an isomorphism \(\rho \) from \({\cal L}\) onto the lattice of subalgebras of \(\frak A^{2}\) with \[\rho(\alpha(x))=(\rho(x))^{-1} (= \{(s,t) : (t,s)\in\rho(x)\})\] for all \(x\in {\cal L}\).

Comment:

Back