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.

9k views

difference between maximal element and greatest element

I know that it's very elementary question but I still don't fully understand difference between maximal element and greatest element. If it's possible, please explain to me this difference with some ...
909 views

518 views

bijective homomorphisms between non isomorphic posets, example , explanation needed

Could someone please give a super simple example of a "bijective homomorphism between a non isomorphic poset"? I don't understand the sentence marked by green in the picture taken from Awodey's book. ...
351 views

With Choice, is any linearly ordered set well-ordered if no subset has order type $\omega^*$?

I've been fumbling around with order types and ordinals these past few days. I read about partial, total, and well-ordered structures, and I'm curious to see if a linearly ordered set has no subset ...
265 views

Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
711 views

Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
243 views

When does a left adjoint between Heyting algebras preserve 1?

This is an exercise from Johnstone's book Stone Spaces: Let $f: A \to B$ and $g: B \to A$ be order-preserving maps between complete Heyting algebras with $f$ left adjoint of $g$. Show that $f$ ...
236 views

Atoms necessary for the existence of a generic filter?

I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me. Definitions We fix some partially ordered set $(P,\preceq)$. A ...
135 views

Intuition for a certain tensor product.

Tensor products occur in lots of places and until recently I thought I understood them at least reasonably well. During the past few weeks, however, I've attended several talks where the tensor ...
324 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact." (The LUB ...
Ordinal exponentiation - $2^{\omega}=\omega$
This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if ...