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Register a new userSolving Equation Systems in $omega$-categorical Algebras
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Thomas Quinn-Gregson
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an $omega$-categorical algebra $mathfrak{A}$. There are $omega$-categorical groups where this problem is undecidable. We show that if $mathfrak{A}$ is an $omega$-categorical semilattice or an abelian group, then the problem is in P or NP-hard. The hard cases are precisely those where Pol$(mathfrak{A}, eq)$ has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras $mathfrak{A}$ such that Pol$(mathfrak{A}, eq)$ does not satisfy this condition, and they are of independent interest in universal algebra. In our proofs we rely on the Barto-Pinsker theorem about the existence of pseudo-Siggers polymorphisms. To the best of our knowledge, this is the first time that the pseudo-Siggers identity has been used to prove a complexity dichotomy.
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This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x) approx m(x,y,x) approx m(x,x,y) approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.
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A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomasse conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number of trees which are sibling of $T$ is either one or infinite. We prove that if $R$ is countable and $aleph_{0}$-categorical, then indeed $sib(R)$ is one or infinite. Furthermore, $sib(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thiery 2013 and studied further by Oudrar and Pouzet 2015, and a result of Frasnay 1984.
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