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We give several new applications of the wreath product of forest algebras to the study of
logics on trees. These include new simplified proofs of necessary conditions for
definability in CTL and first-order logic with the ancestor relation; a
sequence of identities satisfied by all forest languages definable in
PDL; and new examples of languages outside CTL, along
with an application to the question of what properties are definable in both
CTL and LTL.
In an earlier paper, the second author generalized Eilenberg's
variety theory by establishing a basic correspondence between
certain classes of monoid morphisms and families of regular
languages. We extend this theory in several directions. First, we
prove a version of Reiterman's theorem concerning the definition of
varieties by identities, and illustrate this result by describing
the identities associated with languages of the form (a1a2...ak)+, where a1,...,ak are distinct letters. Next, we
generalize the notions of Mal'cev product, positive varieties, and
polynomial closure. Our results not only extend those already known,
but permit a unified approach of different cases that previously
required separate treatment.
We find the atoms of certain subclasses of varieties of finite semigroups and the corresponding varieties of languages. For example we give a new description of languages whose syntactic monoids are R-trivial and idempotent. We also describe the least variety containing all commutative semigroups and at least one non-commutative semigroup. Finally we extend to varieties of finite semigroups some classical results about semilattice decomposition of semigroups.
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