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[parent] distributivity in po-groups (Definition)

Let $ G$ be a po-group and $ A$ be a set of elements of $ G$. Denote the supremum of elements of $ A$, if it exists, by $ \bigvee A$. Similarly, denote the infimum of elements of $ A$, if it exists, by $ \bigwedge A$. Furthermore, let $ A^{-1}=\lbrace a^{-1}\mid a\in A\rbrace$, and for any $ g\in G$, let $ gA=\lbrace ga\mid a\in A\rbrace$ and $ Ag=\lbrace ag\mid a\in A\rbrace$.

  1. If $ \bigvee A$ exists, so do $ \bigvee gA$ and $ \bigvee Ag$.
  2. If 1. is true, then $ g\bigvee A=\bigvee gA =\bigvee Ag$.
  3. $ \bigvee A$ exists iff $ \bigwedge A^{-1}$ exists; when this is the case, $ \bigwedge A^{-1}=(\bigvee A)^{-1}$.
  4. If $ \bigwedge A$ exists, so do $ \bigwedge gA$, and $ \bigwedge Ag$.
  5. If 4. is true, then $ g\bigwedge A=\bigwedge gA=\bigwedge Ag$.
  6. If 1. is true and $ A=\lbrace a,b\rbrace$, then $ a\wedge b$ exists and is equal to $ a(a\vee b)^{-1}b$.
Proof. Suppose $ \bigvee A$ exists.
  • (1. and 2.) Clearly, for each $ a\in A$, $ a\le \bigvee A$, so that $ ga\le g\bigvee A$, and therefore elements of $ gA$ are bounded from above by $ g\bigvee A$. To show that $ g\bigvee A$ is the least upper bound of elements of $ gA$, suppose $ b$ is the upper bound of elements of $ gA$, that is, $ ga\le b$ for all $ a\in A$, this means that $ a\le g^{-1}b$ for all $ a\in A$. Since $ \bigvee A$ is the least upper bound of the $ a$'s, $ \bigvee A\le g^{-1}b$, so that $ g\bigvee A \le b$. This shows that $ g\bigvee A$ is the supremum of elements of $ gA$; in other words, $ g\bigvee A=\bigvee gA$. Similarly, $ \bigvee Ag$ exists and $ g\bigvee A=\bigvee Ag$ as well.
  • (3.) Write $ c=\bigvee A$. Then $ a\le c$ for each $ a\in A$. This means $ c^{-1}\le a^{-1}$. If $ b\le a^{-1}$ for all $ a\in A$, then $ a\le b^{-1}$ for all $ a\in A$, so that $ c\le b^{-1}$, or $ b\le c^{-1}$. This shows that $ c^{-1}$ is the greatest lower bound of elements of $ A^{-1}$, or $ (\bigvee A)^{-1}=\bigwedge A^{-1}$. The converse is proved likewise.
  • (4. and 5.) This is just the dual of 1. and 2., so the proof is omitted.
  • (6.) If $ A=\lbrace a,b\rbrace$, then $ aA^{-1}b=A$, and the existence of $ \bigwedge A$ is the same as the existence of $ \bigwedge (aA^{-1}b)$, which is the same as the existence of $ a (\bigwedge A^{-1}) b$ by 4 and 5 above. Since $ \bigvee A$ exists, so does $ \bigwedge A^{-1}$, and hence $ a (\bigwedge A^{-1}) b$, by 3 above. Also by 3, we have the equality $ a (\bigwedge A^{-1}) b=a(\bigvee A)^{-1} b$. Putting everything together, we have the result: $ a\wedge b=a(a\vee b)^{-1}b$.
This completes the proof. $ \qedsymbol$

Remark. From the above result, we see that group multiplication distributes over arbitrary joins and meets, if these joins and meets exist.

One can use this result to prove the following: every Dedekind complete po-group is an Archimedean po-group.

Proof. Suppose $ a^n\le b$ for all integers $ n$. Let $ A=\lbrace a^n\mid n\in \mathbb{Z}\rbrace$. Then $ A$ is bounded from above by $ b$ so has least upper bound $ \bigvee A$. Then $ a\bigvee A=\bigvee aA=\bigvee A$, since $ aA=A$. As a result, multiplying both sides by $ (\bigvee A)^{-1}$, we get $ a=e$. $ \qedsymbol$

Remark. The above is a generalization of a famous property of the real numbers: $ \mathbb{R}$ has the Archimedean property.



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Cross-references: Archimedean property, real numbers, property, sides, integers, Dedekind complete, joins, meets, arbitrary joins, distributes over, multiplication, group, completes, equality, proof, converse, greatest lower bound, words, upper bound, least upper bound, bounded from above, iff, infimum, supremum, po-group
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This is version 3 of distributivity in po-groups, born on 2007-05-14, modified 2007-05-14.
Object id is 9379, canonical name is DistributivityInPoGroups.
Accessed 662 times total.

Classification:
AMS MSC: 20F60 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Ordered groups)
 06F15 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered groups)
 06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
 06F05 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered semigroups and monoids)
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