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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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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.
   • CommentAuthorRodMcGuire
   • CommentTimeJan 13th 2015
   • (edited Jan 13th 2015)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI noticed that Todd had just edited [[atom]]. I had been meaning to add something explaining the difference between "atomic" and "atomistic" so I added the following at [[atom#remarks_on_terminology]]. >"Atomic" and "Atomistic" differ for the simple example of the [[divisor lattice]] for some number $n$. The atoms in this lattice are prime numbers while it may also contain [[semi-atom|semi-atoms]] which are powers of primes. This lattice is atomic because any object not the [[bottom]] ($1$) is divisible by a prime. However it is not atomistic but instead uniquely semi-atomistic (every non-bottom object is the product of a unique set of semi-atoms), which is one way of stating the [[fundamental theorem of arithmetic]], also known as the *unique factorization theorem*. EDIT: I improved my nLab edit and reflected it above.

   I noticed that Todd had just edited atom. I had been meaning to add something explaining the difference between “atomic” and “atomistic” so I added the following at atom#remarks_on_terminology.

   “Atomic” and “Atomistic” differ for the simple example of the divisor lattice for some number $n$. The atoms in this lattice are prime numbers while it may also contain semi-atoms which are powers of primes. This lattice is atomic because any object not the bottom ($1$) is divisible by a prime. However it is not atomistic but instead uniquely semi-atomistic (every non-bottom object is the product of a unique set of semi-atoms), which is one way of stating the fundamental theorem of arithmetic, also known as the unique factorization theorem.

   EDIT: I improved my nLab edit and reflected it above.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 13th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI did. I've had a lingering feeling of guilt about an interchange between the two of us about the terminology in this entry (which took place some time ago), and I felt it was time to own up more fully to the fact that the conventions you pointed out then are in fact the most common (by far). I like your edits by the way.

   I did. I’ve had a lingering feeling of guilt about an interchange between the two of us about the terminology in this entry (which took place some time ago), and I felt it was time to own up more fully to the fact that the conventions you pointed out then are in fact the most common (by far). I like your edits by the way.

3. • CommentRowNumber3.
   • CommentAuthorRodMcGuire
   • CommentTimeJan 15th 2015
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've improved slightly my addition at [[atom#remarks_on_terminology]], and created the missing page [[semi-atom]]. I've heard the word "semi-atom" used though I don't know if there are other words for this notion or who prefers what. The semi-atom page is now: >A __semi-atom__ is a generalization of the notion of [[atom]] in a [[bottom]] bounded [[partial order]]. >An object $a$ is a semi-atom when the [[interval]] $[\bot, a]$ is a [[chain]], i.e. any two objects $x$ and $y$ in that interval are [[comparable]] ($x \le y$ or $y \leq x$). >All atoms are semi-atoms and usually the bottom is not considered one. >If $a$ and $b$ are semi-atoms then their [[meet]] exists and we have $a \wedge b \in \{\bot, a, b\}$ >## Examples >The [[divisor lattice]] for some number $n$ contains prime numbers as atoms and may contain powers of primes as semi-atoms that are not atoms.

   I’ve improved slightly my addition at atom#remarks_on_terminology, and created the missing page semi-atom. I’ve heard the word “semi-atom” used though I don’t know if there are other words for this notion or who prefers what.

   The semi-atom page is now:

   A semi-atom is a generalization of the notion of atom in a bottom bounded partial order.

   An object $a$ is a semi-atom when the interval $[\bot, a]$ is a chain, i.e. any two objects $x$ and $y$ in that interval are comparable ($x \le y$ or $y \leq x$).

   All atoms are semi-atoms and usually the bottom is not considered one.

   If $a$ and $b$ are semi-atoms then their meet exists and we have $a \wedge b \in \{\bot, a, b\}$

   Examples

   The divisor lattice for some number $n$ contains prime numbers as atoms and may contain powers of primes as semi-atoms that are not atoms.