arXiv subjects

Markus Steindl

Publications and source records attributed to Markus Steindl.

5 recordsLinked to original sources

arXiv2022

Not all nilpotent monoids are finitely related

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every $4$-nilpotent monoid is finitely related. We also give an example of a $5$-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

math.GR↗

arXiv2017

Constrained Best Linear Unbiased Estimation

The least squares (LS) estimator and the best linear unbiased estimator (BLUE) are two well-studied approaches for the estimation of a deterministic but unknown parameter vector. In many applications it is known that the parameter vector fulfills some constraints, e.g., linear constraints. For such situations the constrained LS estimator, which is a simple extension of the LS estimator, can be employed. In this paper we derive the constrained version of the BLUE. It will turn out that the incorporation of the linear constraints into the derivation of the BLUE is not straight forward as for the constrained LS estimator, but the final expression for the constrained BLUE is closely related to that of the constrained LS estimator.

math.ST↗

arXiv2016

The subpower membership problem for semigroups

Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) asks whether $b$ can be generated by $a_1,\ldots,a_k$. If $S$ is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in $nk$. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 letters or more is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup $S$ embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P; otherwise it is NP-complete.

math.GR↗

arXiv2016

On semigroups with PSPACE-complete subpower membership problem

Fix a finite semigroup $S$ and let $a_1, \ldots, a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1, \ldots, a_k$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the $6$-element Brandt monoid, the full transformation semigroup on $3$ or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\ge 2$.

math.GR↗

arXiv2016

The subpower membership problem for bands

Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1,\ldots,a_k$. For bands (idempotent semigroups), we provide a dichotomy result: if a band $S$ belongs to a certain quasivariety, then $SMP(S)$ is in P; otherwise it is NP-complete. Furthermore we determine the greatest variety of bands all of whose finite members induce a tractable SMP. Finally we present the first example of two finite algebras that generate the same variety and have tractable and NP-complete SMPs, respectively.

math.GR↗