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An alternative (tree-based) semantics for a class of regular expressions
is proposed that assigns a central rôle to the + operator and thus to
nondeterminism and nondeterministic choice. For the new semantics a
consistent and complete axiomatization is obtained from the original
axiomatization of regular expressions by Salomaa and by Kozen by dropping
the idempotence law for + and the distribution law of • over +.
The paper contains two remarks on finite sets of groupoid terms closed under subterms and the application of unifying pairs.
It is shown that every concretizable category can be fully embedded into the category of accessible set functors and natural transformations.
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