On extensions of pseudo-valuations on Hilbert algebras

In Buneag (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(I) = 0 and v(x --> y) greater than or equal to v(y) - v(x) for every x, y is an element of A (v is called a valuation if x = 1 whenever v(x) = 0). In Busneag (Math. Japonica 44(2) (1996) 285) it is proved that every pseudo-valuation (valuation) v induces a pseudo-metric (metric) on A defined by d(v)(x, y) = v(x --> y) + v(y --> x) for every x, y is an element of 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), (C) 2002 Elsevier Science B.V. All rights reserved.