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{\bf Sergei Evdokimov and Ilia Ponomarenko}
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{\bf On Highly Closed Cellular Algebras and Highly Closed Isomorphisms}
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We define and study $m$-closed cellular algebras (coherent configurations) and
$m$-iso\-mor\-phisms of cellular algebras which can be regarded as $m$th
approximations of
Schurian algebras (i.e. the centralizer algebras of permutation
groups) and of strong isomorphisms (i.e. bijections of the point sets
taking one algebra to the other) respectively. If $m=1$ we come to
arbitrary cellular
algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms
preserving the Hadamard multiplication). On the other hand, the algebras which
are $m$-closed for all $m\ge 1$ are exactly Schurian ones whereas the weak
isomorphisms which are $m$-isomorphisms for all $m\ge 1$ are exactly ones induced
by strong isomorphisms.  We show that for any $m$ there exist $m$-closed
algebras on $O(m)$ points which are not Schurian
and $m$-isomorphisms of cellular
algebras on $O(m)$ points which are not induced by strong isomorphisms.
This enables us to find for any~$m$ an edge colored
graph with $O(m)$ vertices satisfying the $m$-vertex condition and having
non-Schurian adjacency algebra. On the other hand, we rediscover and
explain from the algebraic point of view the Cai-F{\"u}rer-Immerman
phenomenon that the $m$-dimensional Weisfeiler-Lehman method fails to
recognize the isomorphism of graphs in an efficient way.

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