Modal operators on bounded residuated l -monoids

Jiří Rachůnek; Dana Šalounová

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 299-311
  • ISSN: 0862-7959

Abstract

top
Bounded residuated lattice ordered monoids ( R ℓ -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 MV -algebras (or, equivalently, GMV -algebras) and pseudo BL -algebras (and so, particularly, MV -algebras and BL -algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on MV -algebras were studied by Harlenderová and Rachůnek (2006) and on bounded commutative R ℓ -monoids in our paper which will apear in Math. Slovaca. Now we generalize modal operators to bounded R ℓ -monoids which need not be commutative and investigate their properties also for further derived algebras.

How to cite

top

Rachůnek, Jiří, and Šalounová, Dana. "Modal operators on bounded residuated $\rm l$-monoids." Mathematica Bohemica 133.3 (2008): 299-311. <http://eudml.org/doc/250536>.

@article{Rachůnek2008,
abstract = {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.},
author = {Rachůnek, Jiří, Šalounová, Dana},
journal = {Mathematica Bohemica},
keywords = {residuated l-monoid; residuated lattice; pseudo $\mathop \{\rm BL\}$-algebra; pseudo $\mathop \{\rm MV\}$-algebra; residuated lattice; pseudo BL-algebra; pseudo MV-algebra},
language = {eng},
number = {3},
pages = {299-311},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Modal operators on bounded residuated $\rm l$-monoids},
url = {http://eudml.org/doc/250536},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Rachůnek, Jiří
AU - Šalounová, Dana
TI - Modal operators on bounded residuated $\rm l$-monoids
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 299
EP - 311
AB - 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.
LA - eng
KW - residuated l-monoid; residuated lattice; pseudo $\mathop {\rm BL}$-algebra; pseudo $\mathop {\rm MV}$-algebra; residuated lattice; pseudo BL-algebra; pseudo MV-algebra
UR - http://eudml.org/doc/250536
ER -

References

top
  1. Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C., Cancelative residuated lattices, Algebra Univers. 50 (2003), 83-106. (2003) MR2026830
  2. Blount, K., Tsinakis, C., 10.1142/S0218196703001511, Int. J. Algebra Comput. 13 (2003), 437-461. (2003) Zbl1048.06010MR2022118DOI10.1142/S0218196703001511
  3. Ceterchi, R., Pseudo-Wajsberg algebras, Multiple Val. Logic 6 (2001), 67-88. (2001) Zbl1013.03074MR1817437
  4. Chang, C. C., 10.1090/S0002-9947-1958-0094302-9, Trans. Amer. Math. Soc. 88 (1958), 467-490. (1958) MR0094302DOI10.1090/S0002-9947-1958-0094302-9
  5. Cignoli, R., D'Ottaviano, I. M. L., Mundici, D., Algebraic Foundation of Many-Valued Reasoning, Kluwer, Dordrecht (2000). (2000) 
  6. Nola, A. Di, Georgescu, G., Iorgulescu, A., Pseudo-BL algebras I, Multiple Val. Logic 8 (2002), 673-714. (2002) Zbl1028.06007MR1948853
  7. Dvurečenskij, A., Rachůnek, J., 10.1007/s00233-005-0545-6, Semigroup Forum 72 (2006), 190-206. (2006) MR2216089DOI10.1007/s00233-005-0545-6
  8. Dvurečenskij, A., Rachůnek, J., On Riečan and Bosbach states for bounded Rl-monoids, Math. Slovaca 56 (2006), 487-500. (2006) MR2293582
  9. Font, J. M., Rodriguez, A. J., Torrens, A., Wajsberg algebras, Stochastica 8 (1984), 5-31. (1984) Zbl0557.03040MR0780136
  10. Galatos, N., Tsinakis, C., 10.1016/j.jalgebra.2004.07.002, J. Algebra 283 (2005), 254-291. (2005) MR2102083DOI10.1016/j.jalgebra.2004.07.002
  11. Georgescu, G., Iorgulescu, A., Pseudo- MV algebras, Multiple Val. Logic 6 (2001), 95-135. (2001) MR1817439
  12. Georgescu, G., Leustean, L., 10.1017/S144678870000851X, J. Austral. Math. Soc. 73 (2002), 127-153. (2002) MR1916313DOI10.1017/S144678870000851X
  13. Hájek, P., Basic fuzzy logic and BL-algebras, Soft. Comput. 2 (1998), 124-128. (1998) 
  14. Hájek, P., Observations on non-commutative fuzzy logic, Soft. Comput. 8 (2003), 39-43. (2003) Zbl1075.03009
  15. Harlenderová, M., Rachůnek, J., Modal operators on MV -algebras, Math. Bohem. 131 (2006), 39-48. (2006) MR2211002
  16. Kovář, T., A general theory of dually residuated lattice ordered monoids, PhD Thesis, Palacký University, Olomouc (1996). (1996) 
  17. Kühr, J., Dually residuated lattice-ordered monoids, PhD Thesis, Palacký University, Olomouc (2003). (2003) Zbl1066.06008MR2070377
  18. Kühr, J., Pseudo BL -algebras and DRl-monoids, Math. Bohem. 128 (2003), 199-208. (2003) MR1995573
  19. Leustean, I., 10.1007/s00153-005-0297-8, Arch. Math. Logic. 45 (2006), 191-213. (2006) Zbl1096.03020MR2209743DOI10.1007/s00153-005-0297-8
  20. Macnab, D. S., 10.1007/BF02483860, Alg. Univ. 12 (1981), 5-29. (1981) Zbl0459.06005MR0608645DOI10.1007/BF02483860
  21. Rachůnek, J., A duality between algebras of basic logic and bounded representable DRl-monoids, Math. Bohem. 26 (2001), 561-569. (2001) MR1970259
  22. Rachůnek, J., 10.1023/A:1021766309509, Czech. Math. J. 52 (2002), 255-273. (2002) MR1905434DOI10.1023/A:1021766309509
  23. Rachůnek, J., Šalounová, D., Modal operators on bounded commutative residuated l-monoids, (to appear) in Math. Slovaca. MR2357828
  24. Rachůnek, J., Šalounová, D., A generalization of local fuzzy structures, Soft. Comput. 11 (2007), 565-571. (2007) Zbl1121.06013
  25. Rachůnek, J., Slezák, V., Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures, Math. Slovaca 56 (2006), 223-233. (2006) Zbl1150.06015MR2229343
  26. Swamy, K. L. N., 10.1007/BF01360284, Math. Ann. 159 (1965), 105-114. (1965) Zbl0138.02104MR0183797DOI10.1007/BF01360284

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.