Perfect residuated lattice ordered monoids

Bounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

[1]  Constantine Tsinakis,et al.  The Structure of Residuated Lattices , 2003, Int. J. Algebra Comput..

[2]  Anatolij Dvurecenskij,et al.  Every Linear Pseudo BL-Algebra Admits a State , 2007, Soft Comput..

[3]  Anatolij Dvurecenskij,et al.  States on Pseudo MV-Algebras , 2001, Stud Logica.

[4]  Anatolij Dvurecenskij,et al.  Probabilistic averaging in bounded commutative residuated l-monoids , 2006, Discret. Math..

[5]  George Georgescu,et al.  Some classes of pseudo-BL algebras , 2002, Journal of the Australian Mathematical Society.

[6]  Lavinia Corina Ciungu,et al.  Classes of residuated lattices , 2006 .

[7]  Jiří Rachůnek,et al.  Negation in bounded commutative DRℓ-monoids , 2006 .

[8]  K. Goodearl.  Partially ordered abelian groups with interpolation , 1986 .

[9]  C. Tsinakis,et al.  Cancellative residuated lattices , 2003 .

[10]  Xiao-hong Zhang,et al.  Pseudo-BL algebras and pseudo-effect algebras , 2008, Fuzzy Sets Syst..

[11]  J. Rachunek,et al.  Prime spectra of non-commutative generalizations of MV-algebras , 2002 .

[12]  Anatolij Dvurečenskij,et al.  On Riečan and Bosbach states for bounded non-commutative $R\ell $-monoids , 2006 .

[13]  George Georgescu,et al.  Bosbach states on fuzzy structures , 2004, Soft Comput..

[14]  Jiří Rachůnek,et al.  A non-commutative generalization of MV-algebras , 2002 .

[15]  Jiří Rachůnek,et al.  Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures , 2006 .

[16]  A. Di Nola,et al.  Perfect GMV-Algebras , 2008 .

[17]  Anatolij Dvurecenskij,et al.  Probabilistic Averaging in Bounded Rℓ-Monoids , 2006 .

[18]  Ioana Leustean,et al.  Non-commutative Łukasiewicz propositional logic , 2006, Arch. Math. Log..

[19]  A. Dvurecenskij,et al.  Pseudoeffect Algebras. I. Basic Properties , 2001 .

[20]  A. Dvurecenskij.  Pseudo MV-algebras are intervals in ℓ-groups , 2002, Journal of the Australian Mathematical Society.

[21]  J. Rachunek,et al.  Classes of fuzzy filters of residuated lattice ordered monoids , 2010 .

[22]  B. Riečan.  On the probability theory on MV algebras , 2000, Soft Comput..

[23]  Anatolij Dvurecenskij,et al.  Good and Bad Infinitesimals, and States on Pseudo MV-algebras , 2004, Order.

[24]  Anatolij Dvurecenskij,et al.  Bounded commutative residuated ℓ-monoids with general comparability and states , 2006, Soft Comput..

[25]  A. Dvurečenskij,et al.  On pseudo MV-algebras , 2001, Soft Comput..

[26]  Ioana Leustean,et al.  Local pseudo MV-algebras , 2001, Soft Comput..

[27]  Jirí Rachunek,et al.  A Generalization of Local Fuzzy Structures , 2007, Soft Comput..

[28]  H. Priestley,et al.  Distributive Lattices , 2004 .

[29]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.