Removed the first bullet point in accordance with discussion above, or rather moved the point it was making to the paragraph that follows, where it belongs.

]]>That first bullet point doesn’t really make sense in the context of the paragraph, which was to define a lattice as an algebraic structure (on bare sets, not posets).

I don’t quite see what you are driving at about the symmetry. Yes, the axioms could interchange $\wedge$ and $\vee$ and you’d get the same thing. So? (Cf. the fact that the notion of lattice is self-dual.) But starting from the algebraic axioms before your edit, the standard partial order $\leq$ is defined through $a \leq b$ iff $a = a \wedge b$, which is equivalent to $b = a \vee b$ by absorption, and this restores the asymmetry.

]]>Added missing axiom. To see that an axiom like that is necessary, just observe that in the former formulation (without the first assumption) the requirements for meet and join were symmetric.

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