Articles of order theory

An apex of a cyclically ordered group

Does it make sense to introduce the new definition? Definition. An element $z: z = – z \ne 0$ of a cyclically ordered group is an $apex$ of the group. It looks like an apex has some interesting properties: Lemma 1 (well-known). An apex is unique. Proof (?). Assuming there are two apexes $x$, $y$, […]

How many possible DAGs are there with $n$ vertices

I am have $n$ vertices and trying to enumerate all possible DAGs $\theta$ over $n$. How many DAGs are there? For example when $n=2$, there are 3 possible DAGs and when $n=3$ I tried the following: $|E|=0$, $|\theta|=1$ $|E|=1$, $|\theta|=6$ $|E|=2, |\theta|=8$ $|E|=3,|\theta|=3$ what is the general formula for counting the number of DAGs with […]

Does the empty set have a supremum or infimum?

This question already has an answer here: Infimum and supremum of the empty set 4 answers

What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite subsets of a (possibly infinite) set. The set of all finite-dimensional vector subspaces of a (possibly infinite-dimensional) vector space. […]

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict Orders

As I understand it, partial orders are binary relations that are: Reflexive Anti-symmetric Transitive An example would be $\subseteq$ for sets And if we add totality to this, we get a total (or linear) order, so a total order is Reflexive (this one is implied by totality, so can be removed from definition) Anti-symmetric Transitive […]

All tree orders are lattice orders?

Say that a set is tree ordered if the downset $\downarrow a =\{b:b\leq a\}$ is linearly ordered for each $a$. In a comment, Keinstein says that such sets are also semi-lattices, provided they are connected. This doesn’t seem true to me. Consider e.x. this tree: a c e \ / \ / b d $b\wedge […]

Do linear continua contain $\mathbb{R}$? Can a nontrivial connected space have only trivial path components?

Consider a connected space $X$ under the order topology, such that $X$ contains at least two elements. Does there exist a subspace of $X$ homeomorphic to $\mathbb{R}$? If there isn’t, then $X$ has only trivial path components.

Lexicographical order – posets vs preorders

I found the following definition for lexicographical ordering on Wikipedia (and similar definitions in other places): Given two partially ordered sets $A$ and $B$, the lexicographical order on the Cartesian product $A \times B$ is defined as $(a,b) \le (a’,b’)$ if and only if $a < a’$ or ($a = a’$ and $b \le b’$). […]

What is so special about Higman's Lemma?

Is there a motivational example of an application of Higman’s Lemma that brings out the true beauty and importance of Higman’s Lemma? What is the thing that made so many people care about it? For an example, I was thinking that given a singleton set $\{1\}$ with the linear order $\preceq$ such that basically $1\preceq1$. […]