# Tagged Questions

Order theory deals with properties of orders, usually partial orders or quasi orders but not only those. Questions about properties of orders, general or particular, may fit into this category, as well as questions about properties of subsets and elements of an ordered set.

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### Q: Trichotomy of order of the real numbers

I am currently reading Terence Tao's "Analysis I" and while progressing through the book, the reader is repeatedly asked to prove trichotomy properties of order for the natural numbers $\mathbb{N}$, ...
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### Is every totally ordered finite dimensional vector space a lexicographic order for some basis?

Let's say we have a finite-dimensional vector space $V$ over a totally ordered field $\mathbb{K}$. Is every choice of totally ordered vector space structure (i.e compatible with the addition and ...
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### Well ordering of the subsets of a given set

For a given set, does there always exists a well-ordering of the set of all its subsets which is stronger than the usual ordering (that is set-theoretic inclusion) of the sets of the subsets of the ...
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### Partial order on the set of ordinal functions

Let $\kappa$ be a regular ordinal. Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the ...
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### Is reverse lexicographic order the same as graded reverse lexicographic order?

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse ...
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### Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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### Order Theory and Lattice Theory Synonymous?

Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?
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### Greatest Galois connection

Let $\mathfrak{A}$, $\mathfrak{B}$ be bounded posets. The main question: Explicitly describe (and prove that it exists) the greatest Galois connection between $\mathfrak{A}$ and $\mathfrak{B}$ (as ...
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### Infimum and supremum of subset of inclusion Power set

I'm having trouble understanding the following exercise: Given $U=\{1,2,3,4\}$ with $A=P(U)$ the power set of the elements of $U$ and $R$ the inclusion relation over $A$. Determine the infimum and ...
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### Looking at direction in complex numbers and vector analysis as a cartesian products of 2 imaginary and real total orders.

Can we abstract the idea of direction into an a cartesian product of two total orders? for example:$T_R = \{a,b,...\}$ and $T_I =\{ai,bi,...\}$ where ${T_R}×{T_I}$ is all possible directions and the ...
### There exists a partial ordering of $\mathbb{R}$ with no uncountable chains or antichains
here's an exercise I'm stuck upon. I'm not sure how to approach this: There exists a partial ordering of $\mathbb{R}$ with no uncountable chains or antichains While looking for some help, I ...