Articles of order theory

Longer chains of adjoints :-)

The first question: Is in true that there exist chains of Galois connections (let’s limit to Galois connections between posets) of arbitrary lengths $n$? $F_0(a) \leq b$ if and only if $a \leq F_1(b)$; …; $F_{n-1}(a) \leq b$ if and only if $a \leq F_n(b)$. Then (as it is well known) $F_i(a) = \min \{ […]

Relation between lattice theory and semilattice theory

When studying inverse-semigroups, idempotent elements play an important role, and semilattices occur naturally as sets of idempotent elements that commute with each other. I have the impression that semilattices are closely related to lattices, but the following remark on wikipedia makes me wonder how well this relation is really understood (or taught): Regrettably, it is […]

isomorphism $\Bbb Q$ to $\Bbb Q \cap (0,1)$

I’ve got a question in my homework: Prove that $\langle \mathbb{Q},< \rangle$ and $\langle \mathbb{Q}\cap (0,1),< \rangle$ are isomorphic. I have tried to find a bijective function without any luck. Can anyone please help? Thanks,

How to sort 4 (or more) values with non-piecewise functions?

This question was inspired by this: Finding a non-piece wise function that gives us the $i$'th largest number. My question is how to do this for four or more values. In other words, given 4 values $a, b, c,$ and $d$, specify functions $order_i(a, b, c, d)$ for $i = 1 $ to $4$ such […]

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is the following. Clearly, ∞ isn’t a cardinal and $\aleph_0$ isn’t an extended real number, so we can’t […]

Meet of two elements of a poset, which is a superset of another poset

Let $\mathfrak{A}$ and $\mathfrak{Z}\subseteq\mathfrak{A}$ be two complete lattices (with $\bigcup$ and $\bigcap$ supremum and infimum), order on which agrees. I will denote $\operatorname{up} a = \{ x\in\mathfrak{Z} \mid x\geq a \}$ for every $a\in\mathfrak{A}$. Let the filtrator $(\mathfrak{A},\mathfrak{Z})$ be filtered, that is the following two equivalent (see my free book) conditions hold: $\forall a,b\in\mathfrak{A}: (\operatorname{up}a […]

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 upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in […]

Universality of $(\mathbb Q,<)$

I have few questions to the proof on Universality of linearly ordered sets. Could anyone advise please? Thank you. Lemma: Suppose $(A,<_{1})$ is a linearly ordered set and $(B,<_{2})$ is a dense linearly ordered set without end points. Assume $F\subseteq A$ and $E\subseteq B$ are finite and $h:F \to E$ is an isomorphism from $(F,<_{1})$ […]

How to use well-ordering to form a “least counterexample derived contradiction” to prove rule for obtaining the remainder when dividing $3^n$ by 13?

By the division with remainder theorem, we know that there exists $q \in \mathbb{Z}$ and $r \in \mathbb{Z}$, where $0 \leq r < 13$, such that: \begin{equation*} 3^n = 13q + r \end{equation*} Simple rule for obtaining remainder when dividing $3^n$ by $13$: it can be shown then, that $r = 3^p$, where $p = […]

detecting cofinality in ordinals

Suppose K is an unbounded well ordered set, and suppose K is minimal in the sense that each proper initial segment of K has strictly smaller cardinality. Suppose J is an unbounded well ordered set of the same cardinality as K. Must there exist a subset M of K, and an order preserving embedding f:M—>J […]