# Tagged Questions

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### Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
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### A problem concerning a partially ordered in $\omega$ and two chains.

Assume that $P=\left\langle\omega, \preceq\right\rangle$ is a partially ordered set such that for each $n \in \omega$ there are two chains $A_n$ an $B_n$ in $P$ such that $n \subset A_n \cup B_n$. ...
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### Finite set with $\sup\{S\} \notin S$

Just a quick question about sets. Can anyone here think of an example of a finite set $S$ where the $\sup\{S\} \notin S$ ? I found this on a past exam paper for college, and am unsure if there ...
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### Prove that for every 2 elements in the set F of all functions from N to N, there's an element in F that's bigger than both

let there be $\ F$ the set of all functions from $\ N \rightarrow N$. K is a relation on F, for every f,g$\in$F , (f,g)$\in$K $\leftrightarrow$ for all $\ n\in N$, $\ f(n)\leq g(n)$ Prove that for ...
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### Show that a particular set is a poset

I would like to know if my understanding of the concept of a poset is correct. From what I've learnt from the class: A poset must be transitive, reflexive, and antisymmetric. Am I right? Therefore, ...
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### Completeness of posets

I think this is rather a simple question in order theory, but if someone could explain it step by step that would be really useful. If we have an arbitrary set, then let's denote the set of all finite ...
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### Uniqueness of meets and joins in posets

Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole. Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
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### Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
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### partially ordered group, positive cone, quotient (exercise)

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 ...
In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...