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Pseudobases in direct powers of an algebra. (English) Zbl 0766.08005

A subset \(P\) of an abstract algebra \(A\) is a pseudobasis if every function from \(P\) into \(A\) extends uniquely to an endomorphism on \(A\). \(A\) is called \(\kappa\)-free if \(A\) has a pseudobasis of cardinality \(\kappa\); \(A\) is minimally free if \(A\) has a pseudobasis. The interest of the author is in the existence of pseudobases in direct powers \(A^ I\) of an algebra \(A\). On the positive side, if \(A\) is a rigid division ring, \(\kappa\) is a cardinal, and there is no measurable cardinal \(\mu\) with \(| A| <\mu \leq\kappa\), then \(A^ I\) is \(\kappa\)-free whenever \(| I|=| A^ \kappa|\); on the negative side, if \(A\) is a rigid division ring and there is a measurable cardinal \(\mu\) with \(| A| <\mu \leq | I |\), then \(A^ I\) is not minimally free.

MSC:

08B20 Free algebras
12E15 Skew fields, division rings
03C05 Equational classes, universal algebra in model theory
08A35 Automorphisms and endomorphisms of algebraic structures
08B25 Products, amalgamated products, and other kinds of limits and colimits
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References:

[1] Paul Bankston and Richard Schutt, On minimally free algebras, Canad. J. Math. 37 (1985), no. 5, 963 – 978. · Zbl 0563.08009 · doi:10.4153/CJM-1985-052-8
[2] Paul Bankston, A note on large minimally free algebras, Algebra Universalis 26 (1989), no. 3, 346 – 350. · Zbl 0693.08005 · doi:10.1007/BF01211841
[3] Paul Bankston, Minimal freeness and commutativity, Algebra Universalis 29 (1992), no. 1, 88 – 108. · Zbl 0784.08003 · doi:10.1007/BF01190758
[4] Paul Bankston and Robert A. McCoy, On the classification of minimally free rings of continuous functions, General topology and applications (Middletown, CT, 1988) Lecture Notes in Pure and Appl. Math., vol. 123, Dekker, New York, 1990, pp. 51 – 58. · Zbl 0693.54014
[5] Paul Bankston and Robert A. McCoy, \?-enrichments of topologies, Topology Appl. 42 (1991), no. 1, 37 – 55. · Zbl 0747.54001 · doi:10.1016/0166-8641(91)90031-G
[6] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454. · Zbl 0013.00105
[7] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973. · Zbl 0276.02032
[8] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. · Zbl 0298.02004
[9] Manfred Dugas, Adolf Mader, and Charles Vinsonhaler, Large \?-rings exist, J. Algebra 108 (1987), no. 1, 88 – 101. · Zbl 0616.20026 · doi:10.1016/0021-8693(87)90123-2
[10] E. Fried and J. Sichler, Homomorphisms of commutative rings with unit element, Pacific J. Math. 45 (1973), 485 – 491. · Zbl 0275.13018
[11] George Grätzer, Universal algebra, 2nd ed., Springer-Verlag, New York-Heidelberg, 1979. · Zbl 0412.08001
[12] I. Kříž and A. Pultr, Large \?-free algebras, Algebra Universalis 21 (1985), no. 1, 46 – 53. · Zbl 0597.08010 · doi:10.1007/BF01187555
[13] Péter Pröhle, Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of fields?, Acta Sci. Math. (Szeged) 50 (1986), no. 1-2, 15 – 38. · Zbl 0621.08005
[14] R. Schutt, (private communication).
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