Valuations on the Algebra of Intervals

Quotient BCI-algebras induced by pseudo-valuations

In this paper, we study pseudo-valuations on a BCI-algebra and obtain some related results. The relation between pseudo-valuations and ideals is investigated. We use a pseudo-metric induced by a pseudovaluation to introduce a congruence relation on a BCI-algebra. We define the quotient algebra induced by this relation and prove that it is also a BCI-algebra and study its properties.

MATRIX VALUATION PSEUDO RING (MVPR) AND AN EXTENSION THEOREM OF MATRIX VALUATION

Let R be a ring and V be a matrix valuation on R. It is shown that, there exists a correspondence between matrix valuations on R and some special subsets ?(MVPR) of the set of all square matrices over R, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. Furthermore, based on Malcolmson’s localization, an alternative proof for t...

The multiplicative structure on polynomial continuous valuations

We introduce a canonical structure of a commutative associative filtered algebra with the unit on polynomial smooth valuations and study its properties. The induced structure on the subalgebra of translation invariant smooth valuations has especially nice properties (it is the structure of the Frobenius algebra). We also present some applications.

Residual Ideals of Maclane Valuations

Let K be a field equipped with a discrete valuation v. In a pioneering work, MacLane determined all valuations on K(x) extending v. His work was recently reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié’s approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials which play a key role in t...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations