I tried finding Structures $(X,+,\cdot)$ where X is a set (either finite or infinite) and $+,\cdot$ are operators, for which the following laws apply:
Let $a,b,c\in X$:
$ a\cdot(b+c)=(a\cdot b)+(a\cdot c)\qquad \text{(Distributivity)}\\ a+(b\cdot c)=(a+ b)\cdot (a+ c)\\ a+b=b+a\qquad \text{(Commutativity)}\\ a\cdot b = b\cdot a\\ a\cdot (b \cdot c)=(a\cdot b)\cdot c\qquad \text{(Associativity)}\\ a+(b+c)=(a+b)+c$
And there should be no $a,b\in X$ for which $\forall c\in X: a*b=a*c$ (meaning I dont want 2 elements acting invariant under the Operators, eliminating trivial answers)
I do not require multiplicative or additive inverse (and identity) elements.
Now my Questions: (1) Is there a classification for these structures? I have not found anything on the internet, except for boolean algebras, for which $X=\{0,1\}$ and $+:=\vee,\; \cdot:=\wedge$ (side note: this is where I got the idea from)
(2) I have found, that for any subset of $\mathbb{R}$, $a+b:=\max\{a,b\}$, $a\cdot b:=\min\{a,b\}$ satisfy these rules. However, I have been unable to prove that these are exactly the operators which fulfill them, nor have I been able to find alternative definitions that do. (In boolean algebra, if $true:=1\in\mathbb{R}$ and $false:=0\in\mathbb{R}$ these Operators fit the common definition of $\vee, \wedge$)
Are there other sets of operators (excluding trivial examples, such as $+=\cdot$) that comply with these rules? If not, how is it proven (Over $X\subset\mathbb{R}$)?
I have been trying this for quite a time, but I seem unable to find a proof. ANy clues/hints are welcome, I am willing to work :)