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.
Associativity of multiplication:
a · (b · c) = (a · b) · c
Commutativity of [notaddition] and multiplication:
a ⨨ b = b ⨨ a and a · b = b · a
[notadditive] and multiplicative identity: two different elements exist in W, 0 and 1, such that
a ⨨ 0 = a and a · 1 = a.
[notadditive] inverses: for every a in W, there exists at least one element b in W called a "[notadditive] inverse" of a such that
a ⨨ b = 0. Because [notaddition] is nonassociative, for every a ≠ 0 in W there may exist more than one or two [nonadditive] inverses such that
a ⨨ b = 0 = a ⨨ c = 0 = a ⨨ d and
b ≠ c ≠ d ≠ b.
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.
Left and right distributivity of multiplication over [notaddition]:
a · (b ⨨ c) = (a · b) ⨨ (a · c) = (b ⨨ c) · a.