Abstract
Let
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most
c
n
d
n
cn^{dn}
orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants
c
,
d
c,d
with
d
<
1
d<1
.
We show that
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
is precisely the class of finite covers of first-order reducts of unary structures, and also that
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
is precisely the class of first-order reducts of finite covers of unary structures.
It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
.
We also show that Thomas’ conjecture holds for
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
: all structures in
K
exp
+
\mathcal{K}_{{\operatorname{exp}}{+}}
have finitely many first-order reducts up to first-order interdefinability.
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