9 problems

Let $S_u$ for $u\in\{0,1,2\}$ be the sets … ordered by $x\preccurlyeq y$ if and only if $x>y$. For a well-ordered set $L$ and an ordinal $\alpha$, let $L[\alpha]$ denote the elemen…

Let the sequences referred to in Conjecture 3 be sequences of natural numbers intended to describe numbers of complexity $3n+u$. Conjecture 3. There should be sequences satisfying…

Let $\mathcal C$ be an age of finite structures, ordered by embeddability, and let $[\omega]^{<\omega}$ denote the collection of finite subsets of $\omega$, ordered by inclusion. A…

Let $G$ be a group acting strongly transitively on a totally ordered set $X$ by order-preserving bijections. Amenability conjecture. There exists an amenable group $G$ admitting su…

Closed-upper-set characterization. An upper set $Z$ of $\Lambda^{\ast}$ is closed if and only if it satisfies the four rules.

Kaplansky's conjecture. These two properties characterize the prime spectrum of a commutative ring.

Let $\Phi$ be the set of labeled Fibonacci trees, and let $\vartriangleleft$ be the partial order on $\Phi$ defined by the subtree relation. For $\mathcal{F},\mathcal{F'}\in\Phi$,…

Let $P=(V,\leq)$ be a finite ordered set, meaning that $V$ is finite and $\leq$ is reflexive, antisymmetric, and transitive. A linear extension of $P$ is a linear ordering of $V$ e…

Let $\alpha$ be a countable chain, and let $\mathbb{B}$ be a set of join-semilattices with least element such that an algebraic lattice $L$ contains no chain of order type…