Notes

Description: Definitions from Diener [1989]

Content:

Definitions from Diener [1989]

Definition:

  1. A type \(\tau\) is a function \(\tau = (K_i)_{i\in I}\) where for each \(i\in I\), \(K_i\) is a set (not necessarily finite). If \(I = \{i\}\) and \(K_i = M\) we write \(\tau = (M)\).
  2. An algebra of type \(\tau\) is an ordered pair \({\bf A} = \left(A,(f_i)_{i\in I}\right)\)such that \((\forall i\in I)(f_i:A^{K_i} \to A)\).
  3. A homomorphism from an algebra \(\bf{A}\) of type \(\tau\) to an algebra \({\bf B}\) of type \(\tau\) is closed if the image of every subalgebra of \({\bf A}\) is a subalgebra of \({\bf B}\).
  4. An algebra is a Peano algebra if
    1. The ranges of the operations are pairwise disjoint.
    2. All of the operations are injective.
    3. The elements which are not values of any operation form a generating set.
    4. \(\frak P(\tau;Z)\) is the Peano algebra of type \(\tau\) with generating set \(Z\).

In Diener [1989] it is shown that for any set \(M\), if \(\tau=(M)\), then the axiom of choice for families of cardinality \(|M|\) (that is, \(C(|M|,\infty)\)) is equivalent to the the statement "Every homomorphism of \(\tau\) algebras is closed."  See also forms [1 AP], [1 AQ], [1 AR], [67 E] and [67 F] and notes Note 32 and Note 50.

Howard-Rubin number: 115

Type: Definitions

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