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On the Structure of Group-Like FL\(_e\)-chains

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Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations (IPMU 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 854))

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Abstract

Hahn’s celebrated embedding theorem asserts that linearly ordered Abelian groups embed in the lexicographic product of real groups [13]. In this paper the partial-lexicographic product construction is introduced, a class of residuated monoids, namely, group-like FL\(_e\)-chains which possess finitely many idempotents are embedded into finite partial-lexicographic products of linearly ordered Abelian groups, that is, Hahn’s theorem is extended to this residuated monoid class. As a side-result, the finite strong standard completeness of the logic \(\mathbf{IUL}^{fp}\) is announced.

S. Jenei—The present scientific contribution is dedicated to the \(650^{th}\) anniversary of the foundation of the University of Pécs, Hungary, and was supported by the GINOP 2.3.2-15-2016-00022 grant.

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Notes

  1. 1.

    Real groups are very specific in the class of residuated lattices.

  2. 2.

    In the more general modern terminology.

  3. 3.

    FL-algebras are also called pointed residuated lattices or pointed residuated lattice-ordered monoids.

  4. 4.

    Integrality means that the unit element of the multiplication is the greatest element of the underlying universe.

  5. 5.

    Also called FL\(_{ew}\)-algebras.

  6. 6.

    He called it reducible.

  7. 7.

    In the sense of Clifford.

  8. 8.

    Divisibility is the dual notion of being naturally ordered. For residuated integral monoids, divisibility is equivalent to the continuity of the semigroup operation in the order topology provided that the underlying chain is densely-ordered.

  9. 9.

    In the sense of Aglianò-Montagna.

  10. 10.

    A common generalization of ordinal sums and direct products.

  11. 11.

    It was proved for a wider class of algebras.

  12. 12.

    Absorbent continuity is a weakened version of the naturally ordered property.

  13. 13.

    Weakly real chains are densely-ordered chains with two additional properties.

  14. 14.

    A stronger statement was proved there for at most countable algebras. However, the part of the proof, which is about the densification step does not use this assumption.

  15. 15.

    Or its dual notion, divisibility.

  16. 16.

    That is, all elements are comparable with the unit element of the monoidal operation.

  17. 17.

    Sometimes the lattice operators are replaced by their induced ordering \(\le \) in the signature, in particular, if an FL\(_e\)-chain is considered, that is, if the ordering is linear.

  18. 18.

    That is, there exists a binary operation such that if and only if ; this equivalence is called residuation condition or adjointness condition, () is called an adjoint pair. Equivalently, for any x, z, the set has its greatest element, and is defined as this element: .

  19. 19.

    We use the word monoid to mean semigroup with unit element.

  20. 20.

    Lattice-ordered Abelian groups equipped with and \(f:=t\) are group-like FL\(_e\)-algebras.

  21. 21.

    We set \(\max (\emptyset )=t\).

  22. 22.

    We mean that \((X\setminus X_1)*(X\setminus X_1)\subseteq X\setminus X_1\) holds.

  23. 23.

    We remark that the only choice for \(\mathbf X_1\) is \(\mathbf X_{\tau =t}\), see Definition 5.

  24. 24.

    We mean that for \(x\in X_1\), it holds true that \(x\notin \{ x_\uparrow ,x_\downarrow \}\subset X_1\) (\(\downarrow \) and \(\uparrow \) are computed in X).

  25. 25.

    Just like at item (0), the only choice for \(\mathbf X_1\) is \(\mathbf X_{\tau =t}\), provided that it is discrete, see Definition 5.

  26. 26.

    The rank of an involutive FL\(_e\)-algebra is positive if \(t>f\), negative if \(t<f\), and 0 if \(t=f\).

  27. 27.

    In the spirit of Theorem 1 we identify linearly ordered Abelian groups by cancellative, group-like FL\(_e\)-chains here.

  28. 28.

    \(\mathbf R\) extended by \(\top \) and \(\bot \), just like \(\mathbf Y\) in item (1) of Definition 4.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, University of Pécs, Ifjúság u. 6., Pécs, 7624, Hungary

    Sándor Jenei

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  1. Sándor Jenei
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Correspondence to Sándor Jenei .

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Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain

    Jesús Medina

  2. Universidad de Málaga, Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

  3. Universidad de Granada, Granada, Spain

    José Luis Verdegay

  4. Universidad de Granada, Granada, Spain

    David A. Pelta

  5. Universidad de Málaga, Málaga, Málaga, Spain

    Inma P. Cabrera

  6. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Jenei, S. (2018). On the Structure of Group-Like FL\(_e\)-chains. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 854. Springer, Cham. https://doi.org/10.1007/978-3-319-91476-3_21

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  • DOI: https://doi.org/10.1007/978-3-319-91476-3_21

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