The class of commutative dually residuated lattice ordered monoids (-monoids) contains among others Abelian lattice ordered groups, algebras of Hájek’s Basic fuzzy logic and Brouwerian algebras. In the paper, a unary operation of negation in bounded -monoids is introduced, its properties are studied and the sets of regular and dense elements of -monoids are described.
The notions of Noetherian pseudo MV-algebras and Artinian pseudo MV-algebras are introduced and their characterizations are established. Characterizations of them via fuzzy ideals are also given.
We consider algebras determined by all normal identities of basic algebras. For such algebras, we present a representation based on a q-lattice, i.e., the normalization of a lattice.
We consider algebras determined by all normal identities of -algebras, i.e. algebras of many-valued logics. For such algebras, we present a representation based on a normalization of a sectionally involutioned lattice, i.e. a -lattice, and another one based on a normalization of a lattice-ordered group.