# Tagged Questions

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### Question about primitive group actions

In Glass' Partially Ordered Groups Corollary 7.4.4 says: If $G$ is an ordered group and $(G,G)$ is the right regular representation, then $(G,G)$ is primitive if and only if $G$ is ...
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### lowering of a semilattice

In these Lecture Notes the notion of lowering a semilattice is introduced, there it is stated: Sometimes "broken" elements need to be looked at and computed with. Now any semilattice can have an ...
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### Are ideals of an sup semilattice always non-empty?

I am trying to do an exercise from the book A Compendium of Continuous Lattices. Exercise: Let $L$ be a set with a transitive relation, and let $A,B$ be ideals of $L$. (i) $A\cap B$ is an ideal of ...
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### Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
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### How can we define “trivially orthogonal” groups?

In any lattice-ordered group, we say that two elements are orthogonal if their meet is 1. I've been thinking of groups who have only "trivial" orthogonal relations, i.e. $x\perp y\implies x=1$ or ...
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### Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
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### Intuition behind prime subgroups

In any lattice ordered group, we say that a convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that $X\subseteq P$ or $Y\subseteq P$. This is analogous ...
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### Show $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$

Glass' Partially Ordered GroupsLemma 2.3.2 says: Let G be a p.o. group and $g,h\in G$. If $g\vee h$ exists, then $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$ Proof: If $f\leq g^{-1},h^{-1}$ then ...
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### Show that $\langle G^+\rangle=G$ in a directed group

Lemma 2.1.8 of Glass' Partially Ordered Groups states: $G$ is a directed group if and only if $\langle G^+\rangle=G$ (where $G^+=\{x:x\geq1\}$) This doesn't make any sense to me. For example, ...
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### Why are the p-adic integers a linearly ordered group? [duplicate]

In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group. I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
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### Can all non-archimedean groups be written as a product of archimedean groups?

All the non-archimedean groups I know of can be written as the product of archimedean groups. I'm wondering if this is generally true. I think I've found a proof, but I haven't heard this theorem ...
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### Conditions for linearly ordered groups

One standard definition of a linearly ordered group is that the order $\leq$ obeys the law (1) $a\leq b\implies ac\leq bc$. Suppose (1) only holds for positive c. Must it also hold for negative ...
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### Order-preserving isomorphism between $\mathbb{R}^n$ and $\mathbb{R}$

Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is ...
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### Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?

I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway. Does any ring $R$ exist that satisfies the following properties? $R$ is a totally ordered, ...
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### Monoids with positive and negative elements

By a pointed monoid I mean a monoid $M$ together with an absorbing element $0 \in M$ (i.e. $0x=x0=0$). Equivalently, this is a monoid in the monoidal category $(\mathsf{Set}_*,\wedge)$. By a ...
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### How to derive distributivity from Boolean algebra laws

Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws: $a \wedge b = \bot \Leftrightarrow a \le \neg b$ $\neg\neg a =a$ I'd like to ...
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### Archimedean ordered fields and valuations (exercise)

Definitions: Let $(K,\leq)$ be a totally ordered field and $(G,\leq)$ a totally ordered abelian group (written additively). If we denote $\mathbb{Z}a\!=\!\{na;\, n\!\in\!\mathbb{Z}\}$, then $G$ is ...
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### Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: \begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...
Definitions: A partially ordered group or po-group is a po-set $(G,\leq)$, such that $G$ is a group and $\forall x,y,a,b\!\in\!G\!:x\!\leq\!y\Rightarrow axb\!\leq\!ayb$, i.e. a po-set that is a group ...