Naturally Orderd Transformation Semigroups.
For an abelian lattice ordered group let be the system of all compatible convergences on ; this system is a meet semilattice but in general it fails to be a lattice. Let be the convergence on which is generated by the set of all nearly disjoint sequences in , and let be any element of . In the present paper we prove that the join does exist in .
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.
Let be a multiplicative monoid. If is a non-singular ring such that the class of all non-singular -modules is a cover class, then the class of all non-singular -modules is a cover class. These two conditions are equivalent whenever is a well-ordered cancellative monoid such that for all elements with there is such that . For a totally ordered cancellative monoid the equalities and hold, being Goldie’s torsion theory.
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...
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.