### 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 language | English |
---|---|

Pages (from-to) | 701-713 |

Number of pages | 13 |

Journal | Mathematical Structures in Computer Science |

Volume | 7 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jan 1 1997 |

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### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Computer Science Applications

### Cite this

*Mathematical Structures in Computer Science*,

*7*(6), 701-713. https://doi.org/10.1017/S0960129597002442

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

Research output: Contribution to journal › Article

*Mathematical Structures in Computer Science*, vol. 7, no. 6, pp. 701-713. https://doi.org/10.1017/S0960129597002442

}

TY - JOUR

T1 - Varieties generated by languages with poset operations

AU - Bloom, Stephen L.

AU - Ésik, Z.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=12444333973&partnerID=8YFLogxK

U2 - 10.1017/S0960129597002442

DO - 10.1017/S0960129597002442

M3 - Article

AN - SCOPUS:12444333973

VL - 7

SP - 701

EP - 713

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

SN - 0960-1295

IS - 6

ER -