In this paper we investigate prime divisors, -primes and -primes in -lattices. Using them some new characterizations are given for compactly packed lattices. Next, we study Noetherian lattices and Laskerian lattices and characterize Laskerian lattices in terms of compactly packed lattices.
Dually residuated lattice ordered monoids (-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings (-algebras, -algebras) and their non-commutative variants (-algebras, pseudo -algebras). In the paper, lex-extensions and lex-ideals of -monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.
In this paper we prove for an hl-loop an assertion analogous to the result of Jakubík concerning lexicographic products of half linearly ordered groups. We found conditions under which any two lexicographic product decompositions of an hl-loop with a finite number of lexicographic factors have isomorphic refinements.
Bounded commutative residuated lattice ordered monoids (-monoids) are a common generalization of, e.g., -algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative -monoids are investigated.