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Is there a known algebraic structure with two binary operations defined such that one of the operations behaves like multiplication (identity, distributive, commutative, and associative) and the other behaves like an addition (identity and commutative) that is non-associative?

Generally speaking, is there an online index of algebraic structures searchable by features such as those indicated in the description above -or- is this site the best resource we have?

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    $\begingroup$ Your question is basically already answered at math.stackexchange.com/q/829810/29335 . Wikipedia also contains charts like what you describe (tabular monstrosities listing random properties, structures, and whether or not they have the property.) $\endgroup$ – rschwieb Jan 9 at 15:20
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    $\begingroup$ As for your first question, there are nonassociative structures with two binary operations, but what you've written is way too vague to offer much more advice. When you write "addition (identity and commutative)" i have no idea if you mean to deliberately drop the other group properties that addition has, and I would question whether you really would want to list "commutative" among the properties of multiplication. Nonassociative rings would be a good place to start. $\endgroup$ – rschwieb Jan 9 at 15:23
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    $\begingroup$ "Distributivity" isn't a property of an operation. It is a relationship between two operations. $\endgroup$ – rschwieb Jan 9 at 15:24
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    $\begingroup$ What are "element characteristics"?! $\endgroup$ – rschwieb Jan 9 at 15:36
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    $\begingroup$ Clarifications like that should be edited into the question not buried in the comments. $\endgroup$ – rschwieb Jan 9 at 19:10
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Is there a known algebraic structure with two binary operations defined such that one of the operations behaves like multiplication [...] and the other behaves like an addition (identity and commutative) that is non-associative?

I'm sure there are many. One family of examples of commutative nonassociative algebras is given by Jordan algebras.

(Update: Sorry, I didn't realize you resolved the ambiguity to require that addition be nonassociative. Sorry, no ideas about that one. Never heard of that permutation.)

Generally speaking, is there an online index of algebraic structures searchable by features such as those indicated in the description above -or- is this site the best resource we have?

See this related question.

There is a (not very searchable) listing at http://math.chapman.edu/~jipsen/structures/doku.php/ of different types of structures.

You should probably also read the wiki article on algebraic structures. From time to time an table with a "binary operation" axis and a "property" axis with marks explaining what axioms are required appears. Usually they are quickly deleted since many people find them confusing, space consuming, and generally not useful.

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