On extensions of pseudo-valuations on Hilbert algebras
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- 作者：Busneag ; Dumitru
- 关键词：Hilbert algebra ; Hertz algebra ; Deductive system ; Valuation
- 刊名：Discrete Mathematics
- 出版年：2003
- 出版时间：February 28, 2003
- 年：2003
- 卷：263
- 期：1-3
- 页码：11-24
- 全文大小：152 K
文摘
In Busneag (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 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 dv(x,y)=v(x→y)+v(y→x) for every x,yA, under which → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations).
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 dv(x,y)=v(x→y)+v(y→x) for every x,yA, under which → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations).