Based on the Buşneag’s model ([2, 3, 4]), the notion of pseudovaluations (valuations) on a BE-algebra is introduced, and a pseudometric is induced by a pseudo-valuation on BE-algebras. Related properties are investigated, and conditions for a real-valued function on a BE-algebra to be a pseudo-valuation are discussed. Using the notion of (pseudo) valuation, we show that the binary operation in ...

Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R and σ be an automorphism of R. Then we prove the following: 1. If R is a Pseudo-valuation ring, then R[x, δ] is also a Pseudo-valuation ring. 2. If R is a divided ring, then R[x, δ] is also a divided ring. 3. If R is a Pseudo-valuation ring, thenR[x, x−1, σ] is also a Pseudo-valuation ri...

Recall that a commutative ring R is said to be a pseudo-valuation ring if every prime ideal of R is strongly prime. We define a completely pseudovaluation ring. Let R be a ring (not necessarily commutative). We say that R is a completely pseudo-valuation ring if every prime ideal of R is completely prime. With this we prove that if R is a commutative Noetherian ring, which is also an algebra ov...

Let R be a pseudo-valuation domain with associated valuation domain V and I a nonzero proper ideal of R. Let R̂ (resp., V̂ ) be the I-adic (resp., IV -adic) completion of R (resp., V ). We show that R̂ is a pseudo-valuation domain (which may be a field); and that if I 6= I2, then V̂ is the associated valuation domain of R̂. Let R be an SFT globalized pseudo-valuation domain with associated Prüfer do...

The aim of this paper is to introduce the notion of fantastic deductive systems on generalizations of fuzzy structures, and to emphasize their role in the probability theory on these algebras. We give a characterization of commutative pseudo-BE algebras and we generalize an axiom system consisting of four identities to the case of commutative pseudo-BE algebras. We define the fantastic deductiv...

Ayache has recently proved that if R is an integrally closed domain such that each overring of R is treed, then R is a locally pseudo-valuation domain. We investigate the extent to which the analogue (in which one concludes that R is a locally pseudo-valuation ring) holds if R is generalized to a commutative ring. A positive result is obtained if R is an idealization D(+)K where D is an integra...

the decomposition $gamma=bh$ of a group $gamma$ into a subset $b$ and a subgroup $h$ of $gamma$ induces, under general conditions, a group-like structure for $b$, known as a gyrogroup. the famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by t...

Let V be a valuation domain of rank one with quotient field K. We study the set extensions to rational functions K(X) induced by pseudo-convergent sequences K from topological point view, endowing this either Zariski or constructible topology. In particular, we consider two subspaces prescribed breadth pseudo-limit. give some necessary conditions for space metrizable (under topology) in terms v...

The concept of pseudo valuations on hoop-algebras is introduced, and then the relationship between pseudo-valuations filers investigated. Some conditions for a real-valued function to be valuation are provided. Finally, we prove fact that binary operations uniformly continuous based notion valuation.