Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDaniel Luckhardt
   • CommentTimeNov 11th 2018
   • PermaLink
   Author: Daniel Luckhardt
   Format: MarkdownItexAdded 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. <a href="https://ncatlab.org/nlab/revision/diff/lattice/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/lattice/25">v25</a>, <a href="https://ncatlab.org/nlab/show/lattice">current</a>

   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.

   diff, v25, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 11th 2018
   • (edited Nov 11th 2018)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThat 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 12th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRemoved 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. <a href="https://ncatlab.org/nlab/revision/diff/lattice/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/lattice/27">v27</a>, <a href="https://ncatlab.org/nlab/show/lattice">current</a>

   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.

   diff, v27, current