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.
Closure GMV-algebras are introduced as a commutative generalization of closure MV-algebras, which were studied as a natural generalization of topological Boolean algebras.
Ordered prime spectra of Boolean products of bounded DRl-monoids are described by means of their decompositions to the prime spectra of the components.
It is shown that pseudo BL-algebras are categorically equivalent to certain bounded DRl-monoids. Using this result, we obtain some properties of pseudo BL-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo BL-algebras and, in conclusion, we prove that they form a variety.
Dually residuated lattice-ordered monoids (DRl-monoids for short) generalize lattice-ordered groups and include for instance also GMV -algebras (pseudo MV -algebras), a non-commutative extension of MV -algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.