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.

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...

Abstract We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous and valuations: (1) Let ${\mathcal J}$ be the Johnstone’s non-sober dcpo, μ valuation with ( U )=1 for nonempty Scott opens )=0 $U=\emptyset$ . Then, is a that not minimal. (2) Lebesgue measure extends Sorgenfrey line $\mathbb{R}_\ell$ Its restriction open subsets λ. its im...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in [5] is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in [5] is an automorphism of the algebra of valuations. Then an int...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations wi...