A note on large minimally free algebras
type: Item
instance of: scholarly article
title: A note on large minimally free algebras
zbMATH Open document ID: 0693.08005
DOI: 10.1007/BF01211841
author: Paul Bankston
published in: Algebra Universalis
publication date: 1989-01-01T00:00:00Z
review text: Let A be a (finitary) universal algebra. A is called \(\kappa\)-free (\(\kappa\) a cardinal) provided there exists a subset \(X\subseteq A\) of cardinality \(\kappa\) such that every function f: \(X\to A\) extends uniquely to an endomorphism \(\phi\) : \(A\to A\). So 0-free is the same as rigid. \textit{I. Kříž} and \textit{A. Pultr} [Algebra Univ. 21, 46-53 (1985; Zbl 0597.08010)] have shown that every variety containing arbitrarily large rigid algebras must also contain arbitrarily large \(\kappa\)-free algebras for any fixed \(\kappa\). The author provides a short and elegant proof of this result (in fact, of a more general theorem) by a clever modification of Birkhoff's well-known construction of the free algebra on \(\kappa\) generators relative to the variety generated by A. This approach is completely different from that of Kříž and Pultr.
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 08A35
zbMATH DE Number: 4134124
zbMATH Keywords: universal algebra
zbMATH Keywords: variety
zbMATH Keywords: rigid algebras
zbMATH Keywords: \(\kappa\)-free algebras
MaRDI profile type: Publication
cites work: On minimally Free Algebras
cites work: Q3468288
cites work: Large E-rings exist
cites work: Homomorphisms of commutative rings with unit element
cites work: How comprehensive is the category of semigroups?
cites work: Large k-free algebras
cites work: Varieties of groups
cites work: Q3757954
cites work: Q5555010
description: scientific article; zbMATH DE number 4134124