Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

Wojciech Dzik; Sándor Radeleczki

Bulletin of the Section of Logic (2016)

  • Volume: 45, Issue: 3/4
  • ISSN: 0138-0680

Abstract

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We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

How to cite

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Wojciech Dzik, and Sándor Radeleczki. "Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras." Bulletin of the Section of Logic 45.3/4 (2016): null. <http://eudml.org/doc/295538>.

@article{WojciechDzik2016,
abstract = {We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.},
author = {Wojciech Dzik, Sándor Radeleczki},
journal = {Bulletin of the Section of Logic},
keywords = {filtering unification; compatible operation; intuitionistic logic; Heyting algebra; residuated lattice},
language = {eng},
number = {3/4},
pages = {null},
title = {Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras},
url = {http://eudml.org/doc/295538},
volume = {45},
year = {2016},
}

TY - JOUR
AU - Wojciech Dzik
AU - Sándor Radeleczki
TI - Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras
JO - Bulletin of the Section of Logic
PY - 2016
VL - 45
IS - 3/4
SP - null
AB - We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.
LA - eng
KW - filtering unification; compatible operation; intuitionistic logic; Heyting algebra; residuated lattice
UR - http://eudml.org/doc/295538
ER -

References

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  1. [1] F. Baader, S. Ghilardi, Unification in Modal and Description Logics, Logic Journal of the IGPL, vol. 19 (6), (2011), pp. 705–730. 
  2. [2] F. Baader, W. Snyder, Unification Theory. in A.Robinson, A.Voronkov, (eds.) Handbook of Automated Reasoning, Elsevier Science Publisher, (2001). 
  3. [3] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition, Burris, S., Sankappanavar, H. P. 
  4. [4] X. Caicedo, R. Cignoli, Algebraic Approach to Intuitionistic Connectives Journal of Symbolic Logic 66 (2001), pp. 1620–1636. 
  5. [5] J. L. Castiglioni, M. Menni and M. Sagastume, Compatible operations on commutative residuated lattices, Journal of Applied Non-Classical Logics 18 (2008), pp. 413–425. 
  6. [6] J. L. Castiglioni, H. J. San Martn, Compatible Operations on Residuated Lattices, Studia Logica 98 (2011), pp. 203–222. 
  7. [7] P. Cintula, G. Metcalfe, Admissible rules in the implicationnegation fragment of intuitionistic logic, Annals of Pure and Applied Logic 162, 2 (2010), pp. 162–171. 
  8. [8] W. Dzik, Splittings of Lattices of Theories and Unification Types, Contributions to General Algebra 17 (2006), pp. 71–81. 
  9. [9] W. Dzik, S. Radeleczki, Direct Product of l-algebras and Unification. An Application to Residuated Lattices, Journal of Multiple-valued Logic and Soft Computing 28(2-3) (2017), pp. 189–215. 
  10. [10] L. Esakia, The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic, Journal of Applied Non-Classical Logics 16, No. 3–4 (2006), pp. 349–366. 
  11. [11] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices. An Algebraic Glimpse at Substructural Logics, Studies in logic and the foundations of mathematics, Vol 151, Amsterdam: Elsevier, 2007. 
  12. [12] S. Ghilardi, Unification through Projectivity, Journal of Symbolic Computation 7 (1997), pp. 733–752. 
  13. [13] S. Ghilardi, Unification in Intuitionistic Logic, Journal of Symbolic Logic 64(2) (1999), pp. 859–880. 
  14. [14] S. Ghilardi, L. Sacchetti, Filtering Unification and Most General Unifiers in Modal Logic, Journal of Symbolic Logic 69 (2004), pp. 879–906. 
  15. [15] S. Ghilardi, Best Solving Modal Equations, Annals of Pure and Applied Logic 102 (2000), pp. 183–198. 
  16. [16] T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at logics without contraction, (2001), manuscript. 
  17. [17] A. V. Kuznetsov, On intuitionistic propositional provability calculus, Soviet Math. Doklady 32 (1985), pp. 27–30. 
  18. [18] A. Wronski, On factoring by compact congruences in algebras of certain verieties related to the intuitionistic logic, Bulletin of the Section of Logic 28 (1986), pp. 48–50. 

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