Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
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The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an