# Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups

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DOI: 10.1007/s00224-005-1279-2

- Cite this article as:
- Klima, O., Tesson, P. & Therien, D. Theory Comput Syst (2007) 40: 263. doi:10.1007/s00224-005-1279-2

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## Abstract

We consider the problem of testing whether a given system of equations
over a fixed finite semigroup S has a solution. For the case where
S is a monoid, we prove that the problem is computable in polynomial
time when S is commutative and is the union of its subgroups
but is NP-complete otherwise. When S is a monoid or a
regular semigroup, we obtain similar dichotomies for the restricted
version of the problem where no variable occurs on the right-hand side
of each equation. We stress connections between these problems and constraint
satisfaction problems. In particular, for any finite domain D and
any finite set of relations Γ over D, we construct a finite
semigroup S_{Γ} such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over S_{Γ}.