Two new algebraic equivalents to the axiom of choice
A type \(\tau\) for algebras is a set-valued function \(\tau =(K_ i)_{i\in I}\) on a set I. An algebra of type \(\tau\) is an ordered pair \({\mathfrak A}=<A;(f_ i)_{i\in I}>\), where \((f_ i)_{i\in I}\) is an I- family of \(``K_ i\)-ary'' operations \(f_ i\), that is, \(f_ i: A^{K_ i}\to A\). If all \(K_ i\) are finite, then \({\mathfrak A}\) is said to be finitary. If M is a nonempty set, then the axiom of choice for M- families, \(AC_ M\), asserts that, for every m-family \((N_ m)_{m\in M}\) of nonempty sets, there exists a choice function h: \(M\to \cup_{m\in M}N_ m\) with \(h(m)\in N_ m\) for every \(m\in M.\) Theorem 1.2: If M is a nonempty set and \(\tau =(M)\) is the type consisting of just the ``arity'' M, then \(AC_ M\) is equivalent to the assertion that every homomorphism of a \(\tau\)-algebra \({\mathfrak A}\) into a \(\tau\)-algebra \({\mathfrak B}\) maps subalgebras of \({\mathfrak A}\) onto subalgebras of \({\mathfrak B}.\) The axiom of multiple choice MC asserts that, for every set T and every family \((N_ t)_{t\in T}\) of nonempty sets, there exists a function \(\Psi\) on T such that \(\Psi\) (t) is a nonempty finite subset of \(N_ t\) for every \(t\in T.\) Theorem 2.2: MC is equivalent to the assertion that, for any finitary algebra \({\mathfrak A}=<A;(f_ i)_{i\in I}>\) and any generating subset M of \({\mathfrak A}\), there exists a function h: \(A\to P_{\omega}(M)\) with \(a\in Sg^{{\mathfrak A}}(h(a))\) for every \(a\in A\). (Here, \(Sg^{{\mathfrak A}}(h(a))\) denotes the subalgebra generated by h(a).) Some additional interesting consistency and independence results are too complex to be summarized here.
scientific article; zbMATH DE number 4104943