# Questions tagged [order-theory]

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. Order theory is not about the order of a group nor the order of an element of a group or other algebraic structures.

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### What would this Hasse diagram look like?

Example 8.5.11 in Discrete Mathematics with Applications 5th Edition (Epps) shows finding a topological sorting for a set on the divides relation. (The complete example is shown below) My question : ...
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### implications of pairs of elements generating Boolean subalgebras

I've read about two results which "are well-known", but I haven't found a proof and I haven't been able to prove them by myself yet. So I'd be thankful if someone could give me a hint where ...
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### Relationship between two definitions of being a boolean subalgebra

I've come across two different definitions for subsets to be Boolean, so I'd appreciate if one could tell me if and how these are related: We call an orthocomplemented partially ordered set Boolean, ...
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### Explicit map from countable subset of $[0,1]\subset\mathbb{R}$ to $[0,1]\subset\mathbb{Q}$

Let $A\subsetneq[0,1]$ be some countable set of real numbers. Since the rationals are dense in the reals and since all countable linear orders are embeddable into $\mathbb{Q}$, it seems to me that ...
2answers
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### When does downward closure commute with supremum?

Let $A$ be a suplattices, and suppose we have a family $\{a_i\}_{i\in I}\subseteq A.$ Is $\bigcup_{i\in I}(\operatorname{\downarrow}a_i) = \operatorname{\downarrow} \sup_{i\in I}(a_i)$ in general? ...
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### Can every 2 player game be represented as a sum of rock-paper-scissors games and seed games?

I'm currently trying to formulate the following sentence rigorously: "Given any deterministic two player game (a game such that if two players play multiple times, the result is the same every ...
1answer
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### Equivalent definitions for trees (as partial orders)

Definition. A tree is a partial order $(T,\le)$ which has a least element, and is such that for every $x\in T$, the set $$\downarrow(x):=\{y\in T\mid y\le x\}$$ is well-ordered by the relation $\le$. ...
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### Right adjoint of the inclusion of preorders into small categories

Let $\mathrm{Pre}$ denote the category of preorders, and $\mathrm{Cat}$ the category of small categories. Since every preorder is a category, and monotone map of preorders is a functor, we have the ...
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### every collection $F = \{S_1, . . . , S_n\}$ of $n$ sets contains a sub-collection $S \subseteq F$ of at least $\sqrt{n}$ sets which is union-free

A family of sets $S = \{S_1, \ldots, S_m\}$ is union-free if $S_i \cup S_j \neq S_k$ for all $S_i, S_j , S_k \in S$. Show that every collection $F = \{S_1, \ldots , S_n\}$ of $n$ sets contains a sub-...
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