Buşneag, Dumitru On extensions of pseudo-valuations on Hilbert algebras. (English) Zbl 1018.03050 Discrete Math. 263, No. 1-3, 11-24 (2003). A Hilbert algebra \((A,\rightarrow,1)\) is the algebraic counterpart of Hilbert and Bernays’ positive implicative propositional calculus. A pseudo-valuation (valuation) on \(A\) is a real-valued function \(v\) defined on \(A\) such that \(v(x\rightarrow y)\geq v(y)-v(x)\) and \(v(1)=0\) (and \(v(x)=0\Leftrightarrow x=1\)). The problems studied in this paper can be given the following abstract formulation. Associate with each Hilbert algebra \(A\) a Hilbert algebra \(A^1\) and a map \(j:A\rightarrow A^1\); under what conditions is it true that a pseudo-valuation/valuation \(v\) on \(A\) can be extended to a pseudo-valuation/valuation \(v'\) on \(A^1\) (which means that \(v'\circ j=v\))? The problem is solved in the following cases: 1) \(A^1=A/D\), where \(D\) is a deductive system; 2) \(A^1=Ds(A)\), the dual algebra of deductive systems of \(A\); 3) \(A^1=H_A\), the free Hertz algebra over \(A\); 4) \(A 1=A[S]\), the algebra of fractions relative to a join-closed system \(S\). The paper is remarkably self-contained. Reviewer: Sergiu Rudeanu (Bucureşti) Cited in 2 ReviewsCited in 9 Documents MSC: 03G25 Other algebras related to logic 06D99 Distributive lattices Keywords:Hilbert algebra; Hertz algebra; deductive system; valuation PDFBibTeX XMLCite \textit{D. Buşneag}, Discrete Math. 263, No. 1--3, 11--24 (2003; Zbl 1018.03050) Full Text: DOI