Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
residuated {l}-monoid; residuated lattice; pseudo $\mathop{\rm BL}$-algebra; pseudo $\mathop{\rm MV}$-algebra
residuated {l}-monoid; residuated lattice; pseudo $\mathop{\rm BL}$-algebra; pseudo $\mathop{\rm MV}$-algebra
Summary:
Bounded residuated lattice ordered monoids (${\rm R\ell}$-monoids) form a class of algebras which contains the class of Heyting algebras, i.e.\ algebras of the propositional intuitionistic logic, as well as the classes of algebras of important propositional fuzzy logics such as pseudo $\mathop{\rm MV}$-algebras (or, equivalently, $\mathop{\rm GMV}$-algebras) and pseudo $\mathop{\rm BL}$-algebras (and so, particularly, $\mathop{\rm MV}$-algebras and $\mathop{\rm BL}$-algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on $\mathop{\rm MV}$-algebras were studied by Harlenderová and Rachůnek (2006) and on bounded commutative ${\rm R\ell}$-monoids in our paper which will apear in Math. Slovaca. Now we generalize modal operators to bounded ${\rm R\ell}$-monoids which need not be commutative and investigate their properties also for further derived algebras.
References:
[1] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancelative residuated lattices. Algebra Univers. 50 (2003), 83-106. MR 2026830
[2] Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13 (2003), 437-461. DOI 10.1142/S0218196703001511 | MR 2022118 | Zbl 1048.06010
[3] Ceterchi, R.: Pseudo-Wajsberg algebras. Multiple Val. Logic 6 (2001), 67-88. MR 1817437 | Zbl 1013.03074
[4] Chang, C. C.: Algebraic analysis of many valued logic. Trans. Amer. Math. Soc. 88 (1958), 467-490. DOI 10.1090/S0002-9947-1958-0094302-9 | MR 0094302
[5] Cignoli, R., D'Ottaviano, I. M. L., Mundici, D.: Algebraic Foundation of Many-Valued Reasoning. Kluwer, Dordrecht (2000).
[6] Nola, A. Di, Georgescu, G., Iorgulescu, A.: Pseudo-BL algebras I. Multiple Val. Logic 8 (2002), 673-714. MR 1948853 | Zbl 1028.06007
[7] Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded Rl-monoids. Semigroup Forum 72 (2006), 190-206. DOI 10.1007/s00233-005-0545-6 | MR 2216089
[8] Dvurečenskij, A., Rachůnek, J.: On Riečan and Bosbach states for bounded Rl-monoids. Math. Slovaca 56 (2006), 487-500. MR 2293582
[9] Font, J. M., Rodriguez, A. J., Torrens, A.: Wajsberg algebras. Stochastica 8 (1984), 5-31. MR 0780136 | Zbl 0557.03040
[10] Galatos, N., Tsinakis, C.: Generalized $\rm MV$-algebras. J. Algebra 283 (2005), 254-291. DOI 10.1016/j.jalgebra.2004.07.002 | MR 2102083
[11] Georgescu, G., Iorgulescu, A.: Pseudo-$\rm MV$ algebras. Multiple Val. Logic 6 (2001), 95-135. MR 1817439
[12] Georgescu, G., Leustean, L.: Some classes of pseudo-$\rm BL$ algebras. J. Austral. Math. Soc. 73 (2002), 127-153. DOI 10.1017/S144678870000851X | MR 1916313
[13] Hájek, P.: Basic fuzzy logic and BL-algebras. Soft. Comput. 2 (1998), 124-128.
[14] Hájek, P.: Observations on non-commutative fuzzy logic. Soft. Comput. 8 (2003), 39-43. Zbl 1075.03009
[15] Harlenderová, M., Rachůnek, J.: Modal operators on $\rm MV$-algebras. Math. Bohem. 131 (2006), 39-48. MR 2211002
[16] Kovář, T.: A general theory of dually residuated lattice ordered monoids. PhD Thesis, Palacký University, Olomouc (1996).
[17] Kühr, J.: Dually residuated lattice-ordered monoids. PhD Thesis, Palacký University, Olomouc (2003). MR 2070377 | Zbl 1066.06008
[18] Kühr, J.: Pseudo $\rm BL$-algebras and DRl-monoids. Math. Bohem. 128 (2003), 199-208. MR 1995573
[19] Leustean, I.: Non-commutative Łukasiewicz propositional logic. Arch. Math. Logic. 45 (2006), 191-213. DOI 10.1007/s00153-005-0297-8 | MR 2209743 | Zbl 1096.03020
[20] Macnab, D. S.: Modal operators on Heyting algebras. Alg. Univ. 12 (1981), 5-29. DOI 10.1007/BF02483860 | MR 0608645 | Zbl 0459.06005
[21] Rachůnek, J.: A duality between algebras of basic logic and bounded representable DRl-monoids. Math. Bohem. 26 (2001), 561-569. MR 1970259
[22] Rachůnek, J.: A non-commutative generalization of $\rm MV$-algebras. Czech. Math. J. 52 (2002), 255-273. DOI 10.1023/A:1021766309509 | MR 1905434
[23] Rachůnek, J., Šalounová, D.: Modal operators on bounded commutative residuated l-monoids. (to appear) in Math. Slovaca. MR 2357828
[24] Rachůnek, J., Šalounová, D.: A generalization of local fuzzy structures. Soft. Comput. 11 (2007), 565-571. Zbl 1121.06013
[25] Rachůnek, J., Slezák, V.: Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures. Math. Slovaca 56 (2006), 223-233. MR 2229343 | Zbl 1150.06015
[26] Swamy, K. L. N.: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105-114. DOI 10.1007/BF01360284 | MR 0183797 | Zbl 0138.02104
Search
Partner of
