On extensions of pseudo-valuations on Hilbert algebras Original Research Article

In Buşneag (Math. Japonica 44(2) (1996) 285) I defined a pseudo-valuation on a Hilbert algebra (A,→,1) (cf. (J. Math. 2 (1985) 29; Collection de Logique Math. 21 (1966)) as a real-valued function v on A satisfying v(1)=0 and v(x→y)⩾v(y)−v(x) for every x,y∈A (v is called a valuation if x=1 whenever v(x)=0). In Buşneag (Math. Japonica 44(2) (1996) 285) it is proved that every pseudo-valuation (valuation) v induces a pseudo-metric (metric) on A defined by dv(x,y)=v(x→y)+v(y→x) for every x,y∈A, under which → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations).