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This article indicates that fields may be subclasses of neofields to the extent that they are neofields with associative addition. The author uses the neofield structure exclusively with finite sets, but does not seem to define it with that limitation. Another definition of neofield in this article specifies that the sets of neofields are finite. That would exclude infinite fields with associative addition from being a subclass of neofields.

I've been tinkering with an algebraic structure with infinite sets in mind, and I'd like to know if this may preclude a neofield from being its superclass. While the structure I've tinkered with includes a nonassociative operation, it is by definition not addition, yet has similarities.

Definition of the structure with which I'm tinkering:

A [structure] W is a set with two operations: [notaddition] and multiplication. The result of [notadding] a and b is called the "[notsum]" of a and b. The result of multiplying a and b is called the "product" of a and b. These operations are required to satisfy six properties, referred to as "[structure] axioms." In these axioms, a, b, and c are arbitrary elements of the [structure] W.

  1. Associativity of multiplication:
    a · (b · c) = (a · b) · c

  2. Commutativity of [notaddition] and multiplication:
    ab = ba and a · b = b · a

  3. [notadditive] and multiplicative identity: two different elements exist in W, 0 and 1, such that
    a ⨨ 0 = a and a · 1 = a.

  4. [notadditive] inverses: for every a in W, there exists at least one element b in W called a "[notadditive] inverse" of a such that
    ab = 0. Because [notaddition] is nonassociative, for every a ≠ 0 in W there may exist more than one or two [nonadditive] inverses such that
    ab = 0 = ac = 0 = ad and
    bcdb.

  5. Multiplicative inverses: for every a ≠ 0 in W, there exists a unique element in W denoted by a^−1 called the "multiplicative inverse" of a, such that
    a · a^−1 = 1.

  6. Left and right distributivity of multiplication over [notaddition]:
    a · (bc) = (a · b) ⨨ (a · c) = (bc) · a.

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    $\begingroup$ What would this structure differ from left neofields? $\endgroup$ – Berci Feb 24 at 17:52
  • $\begingroup$ @Berci I updated the expression of the axioms to highlight the potential differences. #6 now includes the specification "left and right" and #4 now makes explicit that "more than one" may also include "more than two." If a left neofield is a loop under "nonassociative addition," my understanding is that precludes it from containing more than two unique inverses for a given element without assuming the cancellation property. I believe that the structure described above may be non-cancellative with regard to that operation. $\endgroup$ – bblohowiak Feb 24 at 21:51
  • $\begingroup$ Any field satisfies your axioms. $\endgroup$ – egreg Feb 24 at 22:25
  • $\begingroup$ @egreg It makes sense that a field would satisfy the axioms; a field is a subclass of neofield and I'm looking for an answer to whether the axioms I offered may characterize a structure that is not a neofield (i.e., both may be superclasses of fields). At this point, I believe a distinction between the two superclasses may hinge upon the possible quantity of unique inverse elements a given element may have under the "nonassociative addition" operation. $\endgroup$ – bblohowiak Feb 25 at 0:41

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