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Are there any clear, accepted examples of operations that are appropriately defined as "addition" but are not associative? Although I can find references to abstract discussions of arithmetic systems with nonassociative addition behind paywalls (Non-Associative Arithmetics, I.M.H. Etherington 1949) or in obscure books (Arithmetic in a Number System with Completely Nonassociative Addition, Mary Bearden Williams 1958) that are not available to me, I cannot find any examples of such operations.

For comparison, a clear example of nonassociative multiplication is the cross product of vectors in three-dimensional Euclidean space.

Machine floating point arithmetic is sometimes posited as an example of nonassociative addition or multiplication, but this seems a rather crude example because the lack of associativity is due to implementation-specific rounding errors, not an intrinsic nonassociativity of adding or multiplying real numbers.

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    $\begingroup$ By "addition," are you referring to the symbol "+" precisely? The symbol is arbitrary, and in mathematics "+" is typically reserved for associative and commutative operations. $\endgroup$
    – davidlowryduda
    Jun 17, 2015 at 14:26
  • $\begingroup$ Addition, multiplication are named given to operations... If you are talking about non associative operations, you already gave good example (vector cross product). If you are talking about addition, the way people use this term, refer to @mixedmath answer. $\endgroup$
    – Martigan
    Jun 17, 2015 at 14:29
  • $\begingroup$ @mixedmath That makes sense...but then what would an entire book about "Arithmetic in a Number System with Completely Nonassociative Addition" be discussing? $\endgroup$ Jun 17, 2015 at 14:42
  • $\begingroup$ @JoshuaHonig What Martigan and mixedmath say is true but there are two other aspects : 1) in rings, fields and their generalizations (i.e. modules, near-ring, near-fields, k-, there are at least two binary operations, and by analogy with the classical situation, one operation is called addition and the other multiplication. 2) addition is typically associated with the idea of the magnitude of the result being close to the magnitude of the larger of the elements being combined and multiplication with expansion of magnitude (analogy to the area, the volume, a list of possibilities etc.) $\endgroup$
    – ogerard
    Jun 20, 2015 at 6:58
  • $\begingroup$ @JoshuaHonig To answer your comment question, I read the thesis by Williams and she gives flesh to the skeletal outline provided by Trevor Evans in his 1957 paper on Nonassociative Number Theory before concluding that any interesting/nontrivial properties of his system lay beyond the scope of common practice. $\endgroup$
    – bblohowiak
    Feb 21, 2019 at 23:34

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As explained in the comments by several contributors, in most standard definitions of mainstream algebraic structures and applications, addition is supposed or defined to be associative (even in near-rings, the first operation is supposed to form a group). It is a useful convention to follow but one can define and study algebras with one, two or more internal binary operations (and even-ary operations) without this restriction but it is wise to use other names.

The two references you found are related to a line of research called "genetic algebra" by some, also "evolution algebra". See MSC 2010 subject 17D92, 17DXX started mainly in the forties by Etherington.

One of the main ideas is the combination/fusion/recombination of two gametes two make one new organism (that's close to an addition, but is non-associative since the order in which your ancestors reproduce matters) as well as the idea of seing familiar numbers as projection of trees. Etherington's name is now associated with various families of trees or equivalently to certain kinds of parenthesizing schemes.

See the Wikipedia entry for basic references on Genetic Algebra.

You might also be interested in other non-classical algebraic structures such as (planar) ternary rings, k-loops, operads but also, even if it is mainly associative, Tropical Mathematics.

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Concatenation can be considered a type of ‘addition’, but concatenation of morphemes is not associative, a counterexample to associativity being the fact that ‘un(peeled)’ and ‘(unpeel)ed’ are antonyms (which is what is behind the fact that ‘unpeeled’ is an auto-antonym).

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Neofields may count. Here is a link to a paper I found that discusses them. Interestingly, it mentions Trevor Evans, the author of the article that the undergraduate work of Mary Bearden Williams built upon.

"The concept of a neofield was first introduced and developed by L.J. Paige in 1949. Paige hoped to use this structure as the co-ordinate system for a finite projective plane and so to construct new planes, poss ibly of non-prime- power order. It turns out, however, that a more natural struc ture to look at is a left neofield because, as was shown by Hsu and Keedwell in a paper published in 1984, left neofields are co-extensive with orthomorphisms and near orthomorphisms of groups. (The present author discovered later that the Ph.D. thesis of C.P. Johnson, a student of the late Trevor Evans, which was submitted in 1981 contains essentially the same idea.)"

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