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On extensions of pseudo-valuations on Hilbert algebras. (English) Zbl 1018.03050

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.

MSC:

03G25 Other algebras related to logic
06D99 Distributive lattices
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