Varieties generated by languages with poset operations

Stephen L. Bloom, Z. Ésik

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 F-labelled posets and any alphabet E we obtain an X-algebra Ex of languages on E. 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 Esik (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 e 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)^ 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 ^ 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.

Original languageEnglish
Pages (from-to)701-713
Number of pages13
JournalMathematical Structures in Computer Science
Volume7
Issue number6
DOIs
Publication statusPublished - Jan 1 1997

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Poset
Algebra
Free Algebras
Subalgebra
Language
Union
Set theory
Binary
Closed
Subset

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Computer Science Applications

Cite this

Varieties generated by languages with poset operations. / Bloom, Stephen L.; Ésik, Z.

In: Mathematical Structures in Computer Science, Vol. 7, No. 6, 01.01.1997, p. 701-713.

Research output: Contribution to journalArticle

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