Let us define an algebraic lattice as a triple $(A,\wedge,\vee)$ in which $\wedge$ and $\vee$ are associative and commutative binary operators which satisfy the laws of absorption: $x{\vee}(x{\wedge}y)=x=x{\wedge}(x{\vee}y)$.
Let us take for granted that we have shown somehow that for $\le$ defined via $x{\le}y\iff x{\wedge}y=x$ the pair $(A,\le)$ is a poset lattice, i.e. it is a poset in which every pair of elements has a supremum and an infimum. I would like to show in addition that the supremum coincides with $\vee$.
Let us start:
- $x\le x{\vee}y$ since $x{\wedge}(x{\vee}y)=x$.
- $y\le x{\vee}y$ since $y{\wedge}(x{\vee}y)=y{\wedge}(y{\vee}x)=y$.
So $x{\vee}y$ is an upper bound of $x$ and $y$. To show that $x{\vee}y$ is the least upper bound of $x$ and $y$, let $z$ be any upper bound of $x$ and $y$. Then $x{\wedge}z=x$ and $y{\wedge}z=y$. We need to show that $(x{\vee}y){\wedge}z$ is equal to $x{\vee}y$.
How? Prove or give a counterexample.
$$x=x\wedge z$$so by absorption$$x\vee z=(x\wedge z)\vee z = z$$and similirly$$y\vee z= z$$ so$$ (x\vee y)\wedge z= (x\vee y)\wedge (x\vee z) = (x\vee y)\wedge (x\vee(y\vee z)) = (x\vee y)\wedge ((x\vee y)\vee z))$$and again, by absorption $$=x\vee y$$