What makes the pairs of operators (-, +) and (/, ×) so similar?
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Short Version When defining the $-$ , $+$ , $/$ , and $×$ operators in a functional manner, one can observe that the $(-, +)$ pair is very similar to the $(/, ×)$ pair, and the only main difference between them is their identity terms ( $0$ and $1$ respectively). My questions are the following: where can I find some prior work on this topic, and can one define a family of such operator pairs with different identity terms? Is there any theory for such objects? Set Theory Version While the arithmetic properties outlined below can be defined for both sets and types, referring to set theory might help clarify the question: if $(+, -)$ with 0 as identity element defines a group and $[(+, -), (×, /)]$ with 1 as identity element for $(×, /)$ defines a field, what is defined by $[(+, -), (×, /), (#, @)]$ wit