Two new algebraic equivalents to the axiom of choice
type: Item
instance of: scholarly article
title: Two new algebraic equivalents to the axiom of choice
zbMATH Open document ID: 0675.03030
DOI: 10.1007/BF01236864
author: Karl-Heinz Diener
published in: Algebra Universalis
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 08A65
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 4104943
zbMATH Keywords: axiom of choice
zbMATH Keywords: axiom of multiple choice
reviewed by: Elliott Mendelson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01236864
OpenAlex ID: W1974552507
Wikidata QID: Q114694005
description: scientific article; zbMATH DE number 4104943