This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So …

The original Kuratowski closure-complement problem asks: How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given s …

Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$). $f$ i …

Let A is a complete lattice. I call a subset $S$ of A filter-closed when for every filter base $T$ in $S$ we have $\bigcap T\in S$. (A filter base is a nonempty, down directed set …

Let $U$ is a complete lattice with least element 0. Weak partitioning is a collection $S$ of nonempty subsets of $U$ such that $\forall x\in S: x\cap\bigcup(S\setminus\{x\})=0$. …

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you w …

Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is ther …

[Added: This is a follow-up of an earlier post.] Consider the following "reconstruction puzzle", stated informally: Given a concrete poset, e.g. the poset of undirected unlabeled …

Given a partial order R≤ over a set D, the set of upper bounds under R of a subset S of D is commonly defined as { y ∈ D | ∀ x ∈ S, x R y }. (The set of lower bounds of S may be d …

A Boolean lattice has a number of rather nice properties which give it a central role in many parts of combinatorics. For instance, it's a lattice, it can be augmented with a ring …

Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? T …

The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itsel …

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) Let ⋄ be the 4 element lattice τ / \ i j \ / f Is every lattice isomorphic to the fixed point la …

(Asked by Nathaniel Hellerstein on the Q&A board at JMM) This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a …

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, wh …