×

zbMATH — the first resource for mathematics

Direct product of \(\ell\)-algebras and unification: an application to residuated lattices. (English) Zbl 1398.08007
Summary: We describe classes of \(\ell\)-algebras (which are based on lattices) such that their finitely presented projective algebras are closed under finite direct product, that is, for which unification is filtering. This implies that unification in such classes is either unitary or nullary. Following ideas of S. Ghilardi [J. Log. Comput. 7, No. 6, 733–752 (1997; Zbl 0894.08004); J. Symb. Log. 69, No. 3, 879–906 (2004; Zbl 1069.03011)] we attempt to describe filtering unification in a variety by means of properties of factor-congruences of algebras of the variety. The results subsume some previous results, but not those of [loc. cit.], and open new areas for applications like residuated lattices. In particular we show that filtering unification depends on the monoid operation, that is, unification is filtering in varieties generated by residuated lattices without zero divisors. This implies that unification in strict fuzzy logics such as SMTL, MTL and many others is unitary or nullary.

MSC:
08B05 Equational logic, Mal’tsev conditions
08A05 Structure theory of algebraic structures
06B20 Varieties of lattices
03G25 Other algebras related to logic
PDF BibTeX XML Cite
Full Text: Link
References:
[1] [1] Baader, F., Ghilardi, S. (2011). Unification in Modal and Description Logics. Logic Journal of the IGPL, Vol. 19, No. 6, 705-730. · Zbl 1258.03018
[2] [2] Baader, F., Snyder, W. (2001). Unification Theory. In: Robinson, A. and Voronkov, A. (eds.) Handbook of Automated Reasoning. Ch.8, Elsevier Science Publ., MIT, 1, 445- 533. · Zbl 1011.68126
[3] [3] Burris, S. (1992). Discriminator Varieties and Symbolic Computation. J. of Symbolic Computation, 13, 175-207. · Zbl 0803.08002
[4] [4] Cs´ak´any, B. (1970). Characterizations of regular varieties. Acta Sci. Math. (Szeged), 31, 187-189.
[5] [5] Cignoli, R., Esteva, F. (2009). Commutative integral bounded residuated lattices with an added involution. Annals of Pure and Applied Logic, 161, 150-160. · Zbl 1181.03061
[6] [6] Dzik, W. (2006). Splittings of Lattices of Theories and Unification Types. Contributions to General Algebra, 17, 71-81. · Zbl 1109.06008
[7] [7] Dzik, W. (2007). Unification types in logic. Katowice: Wydawnictwo Uniwersytetu Slaskiego. · Zbl 1148.03003
[8] [8] Dzik, W. (2008). Unification in Some Substructural Logics of BL-algebras and Hoops. Reports on Mathematical Logic, 43, 73-83. · Zbl 1156.03022
[9] [9] Estava, F., Godo, L. (2001). Monoidal t-norm based logic: Towards a logic of leftcontinous t-norms. Fuzzy Sets and Systems, 124, 271-288.
[10] [10] Estava, F., Godo, L., Garcia-Cerdana, A. (2003). On the Hierarchy of t-norm based Residuated Fuzzy Logic. In: Fitting, M., Orlowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic, Physica-Verlag HD.
[11] [11] Galatos, N., Jipsen, P., Kowalski, T. and Ono, H. (2007). Residuated lattices: an algebraic glimpse at substructural logics. Studies in logic and the foundations of mathematics, Vol. 151, Amsterdam: Elsevier. · Zbl 1171.03001
[12] [12] Ghilardi, S. (1997). Unification through Projectivity. J. of Symbolic Computation, 7 , 733- 752. · Zbl 0894.08004
[13] [13] Ghilardi, S. (2004). Unification, Finite Duality and Projectivity in Locally Finite Varieties of Heyting Algebras. Annals of Pure and Applied Logic, 127, No. 1-3, 99-115. · Zbl 1058.03020
[14] [14] Ghilardi, S., Sacchetti, L. (2004). Filtering Unification and Most General Unifiers in Modal Logic. J. of Symbolic Logic, 69, 879-906. · Zbl 1069.03011
[15] [20] Kowalski, T., Ono, H. (2001). Residuated lattices: An algebraic glimpse at logics without contraction, manuscript.
[16] [21] Maeda, F., Maeda, S. (1970). Theory of Symmetric lattices, Springer-Verlag. · Zbl 0219.06002
[17] [22] Radeleczki, S. (2003). The direct decomposition of-algebras into products of subdirectly irreducible factors. J. Aust. Math. Soc. Ser. A, 75, 41-56. · Zbl 1045.08002
[18] [23] Radeleczki, S. (2000). Classification systems and the decomposition of a lattice in direct products. · Zbl 1062.06008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.