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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 and 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 is defined through iff , which is equivalent to by absorption, and this restores the asymmetry.
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