12 problems

The Tarski alternative. Exactly one of the following is true:

Let a general higher-rank graph be given, and consider the inverse semigroup constructed from it as in Farthing, Muhly, and Yeend (2005). Let $G_\mathrm{tight}$ denote the tight gr…

Classification conjecture. One of the following holds:

Henckell–Rhodes conjecture. Every finite inverse monoid $M$ has a finite $F$-inverse cover: there is an idempotent-separating homomorphism from a finite $F$-inverse monoid onto $M$…

Compactness conjecture. Every Nyikos inverse topological semigroup is compact.

Let two Markov shifts be given, with associated inverse hulls and alphabets. Alphabet-size conjecture. If the inverse hulls of the two Markov shifts are isomorphic, then their asso…

Conjugacy-class cardinality conjecture. The conjugacy class of $e$ has

Let $S$ be an inverse semigroup with zero. Say that $S$ satisfies Condition (K) if and only if $S/I$ satisfies Condition (L) for every saturated ideal $I$ of $S$. The notation…

Few-idempotents conjecture. A noble inverse semigroup must have few idempotents.

Classification conjecture. The monoid $S$ is isomorphic to one of the following:

Finite-order injectivity conjecture. Then

Directed joint enlargement conjecture. If $S$ and $T$ are both directed, then they are Morita equivalent if and only if they have an inverse semigroup joint enlargement.