# Article

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Keywords:
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. 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. 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. 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. 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. 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). 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. MR 2209743 | Zbl 1096.03020
[20] Macnab, D. S.: Modal operators on Heyting algebras. Alg. Univ. 12 (1981), 5-29. 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. 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. MR 0183797 | Zbl 0138.02104

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