The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (Rl-monoids) are common generalizations of BL-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded Rl-monoids.
Dually residuated lattice ordered monoids (DRl-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings (MV -algebras, BL-algebras) and their non-commutative variants (GMV - algebras, pseudo BL-algebras). In the paper, lex-extensions and lex-ideals of DRl-monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.
Bounded commutative residuated lattice ordered monoids ($R\ell $-monoids) are a common generalization of, e.g., $BL$-algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative $R\ell $-monoids are investigated.
Bounded residuated lattice ordered monoids (Rl-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 Rl-monoids in our paper which will apear in Math. Slovaca. Now we generalize modal operators to bounded Rl-monoids which need not be commutative and investigate their properties also for further derived algebras.
Weakly associative lattice rings (wal-rings) are non-transitive generalizations of lattice ordered rings (l-rings). As is known, the class of l-rings which are subdirect products of linearly ordered rings (i.e. the class of f-rings) plays an important role in the theory of l-rings. In the paper, the classes of wal-rings representable as subdirect products of to-rings and ao-rings (both being non-transitive generalizations of the class of f-rings) are characterized and the class of wal-rings having lattice ordered positive cones is described. Moreover, lexicographic products of weakly associative lattice groups are also studied here.