Direct decompositions of dually residuated lattice-ordered monoids

@article{Rachunek2004DirectDO,
  title={Direct decompositions of dually residuated lattice-ordered monoids},
  author={Jir{\'i} Rachunek and Dana Salounov{\'a}},
  journal={Discussiones Mathematicae General Algebra and Applications},
  year={2004},
  volume={24},
  pages={63-74}
}
The class of dually residuated lattice ordered monoids (DR`-monoids) contains, in an appropriate signature, all `-groups, Brouwerian algebras, MV and GMV -algebras, BLand pseudo BL-algebras, etc. In the paper we study direct products and decompositions of DR`monoids in general and we characterize ideals of DR`-monoids which are direct factors. The results are then applicable to all above mentioned special classes of DR`-monoids. 
View via Publisher
discuss.wmie.uz.zgora.pl
3 Citations

3 Citations

Citation Type

Citation Type

LEXICOGRAPHIC EXTENSIONS OF DUALLY RESIDUATED LATTICE ORDERED MONOIDS
Dually residuated lattice ordered monoids (DR`-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings (MV -algebras,…
Direct product decompositions of bounded commutative residuated ℓ-monoids
The notion of bounded commutative residuated ℓ-monoid (BCR ℓ-monoid, in short) generalizes both the notions of MV-algebra and of BL-algebra. Let be a BCR ℓ-monoid; we denote by ℓ() the underlying…
Direct product factors in GMV-algebras
GMV-algebras are non-commutative generalizations of MV-alge­ bras and by A. Dvurecenskij they can be represented as intervals of unital lattice ordered groups. Moreover, they are polynomially…

References

SHOWING 1-10 OF 45 REFERENCES
A DUALITY BETWEEN ALGEBRAS OF BASIC LOGIC AND BOUNDED REPRESENTABLE DRl-MONOIDS
BL-algebras, introduced by P. Hajek, form an algebraic counterpart of the basic fuzzy logic. In the paper it is shown that BL-algebras are the duals of bounded representable DRl-monoids. This duality…
Pseudo $BL$-algebras and $DR\ell $-monoids
It is shown that pseudo $BL$-algebras are categorically equivalent to certain bounded $DR\ell $-monoids. Using this result, we obtain some properties of pseudo $BL$-algebras, in particular, we can…
Ideals in autometrized algebras
Abstract A notion of a normal autometrized algebra is introduced which generalises the concepts of Boolean geometry. Brouwerian geometry, autometrized lattice ordered groups, semi-Brouwerian…
Ideals of noncommutative DRℓ-monoids
In this paper, we introduce the concept of an ideal of a noncommutative dually residuated lattice ordered monoid and we show that congruence relations and certain ideals are in a one-to-one…
Prime spectra of non-commutative generalizations of MV-algebras
Abstract. Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We…
Metamathematics of Fuzzy Logic
  • P. Hájek
  • Philosophy, Computer Science
    Trends in Logic
  • 1998
TLDR
This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.
"J."
however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)…
A general theory of dually residuated lattice-ordered monoids, Ph.D. Thesis, Palacký Univ., Olomouc 1996. Direct decompositions of dually residuated .
  • 1996
Dually Residuated Lattice Ordered Semigroups, III
Lex-ideals of DR$\ell $-monoids and GMV-algebras
...
...