Let our language be $\{+,* \}$. $*$ is said to be left-distributive over $+$ iff $x * (y + z)=(x*y)+(x*z)$, and is said to be right-distributive over $+$ iff $(x+y)*z=(x*z)+(y*z)$. That raises the question, in the language of $\{+,*\}$, is there a single identity which is equivalent to the conjunction of the left and right distributive laws?
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2If you expand the language with a multiplicative identity $1$, then the answer is yes: $$(t(u+v))w=((tu)w)+((tv)w).$$ In general, I don't know. But notice that often, when several identities are merged into one, that one is cumbersome to deal with. – amrsa Jul 23 '21 at 17:25
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1Likewise, an element $0$ such that $x0=0x=0$ would give the result, with $$(t(u+v))+((u+v)w)=((tu)+(tv))+((uw)+(vw)).$$ – amrsa Jul 26 '21 at 08:40