Inversion in a class of lattice-ordered algebras
Lattice effect algebras densely embeddable into complete ones
An effect algebraic partial binary operation defined on the underlying set uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion of there exists an effect algebraic partial binary operation then need not be an extension of . Moreover, for an Archimedean atomic lattice effect algebra we give a necessary and sufficient condition for that existing on is an extension of defined on . Further we show that such extending exists at most...
Lattice-ordered algebras with polynomial inequalities.
Lineare Erweiterungen der teilweisen Anordnung in Ringen
Matrices ordonnables : une étude algébrique
Matrix representable so-rings.
Model-completeness for sheaves of structures
Modules ordonnés injectifs
Non-transitive generalizations of subdirect products of linearly ordered rings
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...
Norms on semirings. I.
Number of polynomials in ordered algebras
On a functional equation arising from hyperbolic geometry.
On algebra homomorphisms in complex almost -algebras
Extensions of order bounded linear operators on an Archimedean vector lattice to its relatively uniform completion are considered and are applied to show that the multiplication in an Archimedean lattice ordered algebra can be extended, in a unique way, to its relatively uniform completion. This is applied to show, among other things, that any order bounded algebra homomorphism on a complex Archimedean almost -algebra is a lattice homomorphism.
On algebra-lattices
On congruence distributivity of ordered algebras with constants
We define the order-congruence distributivity at 0 and order- congruence n-distributivity at 0 of ordered algebras with a nullary operation 0. These notions are generalizations of congruence distributivity and congruence n-distributivity. We prove that a class of ordered algebras with a nullary operation 0 closed under taking subalgebras and direct products is order-congruence distributive at 0 iff it is order-congruence n-distributive at 0. We also characterize such classes by a Mal'tsev condition....
On Conrad's partial order relation on semiprime rings and semigroups.
On extensions of orthosymmetric lattice bimorphisms
In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if is a commutative -algebra and its Dedekind completion, then, can be equipped...
On -rings that are not formally real
Henriksen and Isbell showed in 1962 that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings) that are satisfied by all totally-ordered fields. In this paper, we review the work of Henriksen and Isbell on this topic, construct and classify some examples that illustrate this phenomenon using the valuation theory of Hion (in the process, answering a question posed in [E]) and, finally, prove that a base for the equational theory of totally-ordered...
On ordered division rings
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x ↦ xa² for non-zero a, in place of requiring that positive elements have a positive product. Our aim in this work is to study this type of ordering in the case of a division ring. We show that it actually behaves just as in the commutative...
On ordered division rings
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under for nonzero , instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...