Canonicity in subvarieties of BL-algebras. (English) Zbl 1200.03049
Canonical extensions were introduced by Jonsson and Tarski for Boolean algebras with operators and generalized for distributive lattices, lattices and posets with different internal operations. They provide an algebraic formulation of what is otherwise treated via topological duality or relational methods.
In this paper, the authors study \(\sigma\)- and \(\pi\)-canonicity for subvarieties of BL-algebras.
They prove that every subvariety of BL-algebras that is not finitely generated is not \(\sigma\)-canonical. Also, they prove \(\pi\)-canonicity for an infinite family of subvarieties of BL-algebras that are not finitely generated.
In this paper, the authors study \(\sigma\)- and \(\pi\)-canonicity for subvarieties of BL-algebras.
They prove that every subvariety of BL-algebras that is not finitely generated is not \(\sigma\)-canonical. Also, they prove \(\pi\)-canonicity for an infinite family of subvarieties of BL-algebras that are not finitely generated.
Reviewer: Dana Piciu (Craiova)
MSC:
| 03G25 | Other algebras related to logic |
References:
| [1] | Agliano R., Montagna F.: Varieties of BL-algebras I: general properties. Journal of Pure and Applied Algebra 181, 105–129 (2003) · Zbl 1034.06009 · doi:10.1016/S0022-4049(02)00329-8 |
| [2] | Blok W.J., Ferreirim I.M.A.: On the structure of hoops. Algebra Universalis 43, 233–257 (2000) · Zbl 1012.06016 · doi:10.1007/s000120050156 |
| [3] | Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Springer, New York (1981) · Zbl 0478.08001 |
| [4] | Busaniche M.: Free algebras in varieties of BL-algebras generated by a chain. Algebra Universalis 50, 259–277 (2003) · Zbl 1081.03063 · doi:10.1007/s00012-003-1835-z |
| [5] | Busaniche M.: Decomposition of BL-chains. Algebra Universalis 52, 519–525 (2004) · Zbl 1077.03043 · doi:10.1007/s00012-004-1899-4 |
| [6] | Busaniche M., Cignoli R.: Free algebras in varieties of BL-algebras generated by a BL n -chain. Journal of the Australian Mathematical Society 80, 419–439 (2006) · Zbl 1094.03058 · doi:10.1017/S1446788700014117 |
| [7] | Cabrer L.M.: Non canonicity of BL-algebras. Reports on Mathematical Logic 44, 107–125 (2008) |
| [8] | Cignoli, R., D’Ottaviano, M.I., Mundici, D.: Algebraic foundations of many-valued reasoning. Kluwer Academic Pub., Dordrecht (2000) · Zbl 0937.06009 |
| [9] | Cignoli R., Torrens A.: An algebraic analysis of product logic. Multiple Valued Logics 5, 45–65 (2000) · Zbl 0962.03059 |
| [10] | Dummett M.: A propositional calculus with denumerable matrix. J. Symb. Log. 24, 97–106 (1959) · Zbl 0089.24307 · doi:10.2307/2964753 |
| [11] | Dunn M., Gehrke M., Palmigiano A.: Canonical extensions and relational completness of some substructural logics. Journal of Symbolic Logic 70, 713–740 (2005) · Zbl 1101.03021 · doi:10.2178/jsl/1122038911 |
| [12] | Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logics and the Foundations of Mathematics, vol. 151. Elsevier (2007) · Zbl 1171.03001 |
| [13] | Gehrke M., Harding J.: Bounded lattice expansions. Journal of Algebra 238, 345–371 (2001) · Zbl 0988.06003 · doi:10.1006/jabr.2000.8622 |
| [14] | Gehrke M., Jónsson B.: Bounded distributive lattices with operators. Mathematica Japonica 40, 207–215 (1994) · Zbl 0855.06009 |
| [15] | Gehrke M., Jónsson B.: Bounded distributive lattice expansions. Mathematica Scandinavica 94, 13–45 (2004) · Zbl 1077.06008 |
| [16] | Gehrke M., Priestley H.: Non-canonicity of MV-algebras. Houston Journal of Mathematics 28, 449–455 (2002) · Zbl 1060.06019 |
| [17] | Hájek P.: Metamathematics of Fuzzy Logic. Kluwer Academic Pub., Dordrecht (1998) · Zbl 0937.03030 |
| [18] | Hájek P.: Basic fuzzy logic and BL-algebras. Soft Computing 2, 124–128 (1998) |
| [19] | Horn A.: Logic with truth values in a linearly ordered Heyting algebra. J. Symb. Log. 34, 395–408 (1969) · Zbl 0181.29904 · doi:10.2307/2270905 |
| [20] | Jónsson B., Tarski A.: Boolean algebras with operators. American Journal of Mathematics 73, 891–939 (1951) · Zbl 0045.31505 · doi:10.2307/2372123 |
| [21] | Jónsson B., Tarski A.: Boolean algebras with operators II. American Journal of Mathematics 74, 127–162 (1952) · doi:10.2307/2372074 |
| [22] | Kowalski T., Litak T.: Completions of GBL-algebras: negative results. Algebra Universalis 58, 373–384 (2008) · Zbl 1161.06010 · doi:10.1007/s00012-008-2056-2 |
| [23] | Monteiro A. (1962) Linéarisation de la logique positive de Hilbert Bernays. Revista Un. Mat. Argentina 20, 308–309 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
