Author:
BLOOM STEPHEN L.,ÉSIK ZOLTÁN
Abstract
A V-labelled poset P can induce an operation
on
the languages on any fixed alphabet, as
well as an operation on labelled posets (as noticed by Pratt and Gischer
(Pratt 1986; Gischer 1988)). For any collection X of
V-labelled posets and any alphabet Σ we obtain an
X-algebra ΣX of languages on Σ.
We
consider the variety Lang(X) generated by these
algebras when X is a collection of nonempty ‘traceable
posets’. The current paper contains
several observations about this variety. First, we use one of the
basic results in Bloom and
Ésik (1996) to show that a concrete description of the
A-generated free algebra in Lang(X)
is the X-subalgebra generated by the singletons (labelled
a∈A) in the X-algebra of all
A-labelled posets. Equipped with an appropriate ordering, these
same
algebras are the free ordered algebras in the variety
Lang(X)[les ] of ordered language X-algebras.
Further, if one enriches the language algebras by adding either a binary
or
infinitary union operation, the
free algebras in the resulting variety are described by certain
‘closed’ subsets of the original
free algebras. Second, we show that for ‘reasonable sets’ X,
the variety Lang(X) has the
property that for each n[ges ]2, the n-generated free
algebra is a subalgebra of the 1-generated
free algebra. Third, knowing the free algebras enables us to show that
these varieties are
generated by the finite languages on a two-letter alphabet.
Publisher
Cambridge University Press (CUP)
Subject
Computer Science Applications,Mathematics (miscellaneous)
Cited by
5 articles.
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