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Algebraic properties of pre-logics. (English) Zbl 1007.08003
A pre-logic is an algebra \((A,\cdot ,1)\) of type \((2,1)\) satisfying the identities \(xx=1\), \(1x=x\), \(x(yz)=(xy)(xz)\) and \(x(yz)=y(xz)\). Pre-logics generalize Hilbert algebras which model the operation of implication in intuitionistic logic. Let \(\mathcal A=(A,\cdot ,1)\) be a pre-logic. \(D\subseteq A\) is called a deductive system of \(\mathcal A\) if \(1\in D\) and if \(x\in D\), \(y\in A\) and \(xy\in D\) imply \(y\in D\). A non-empty subset \(I\) of \(A\) is called an ideal of \(\mathcal A\) if \(x\in A\) and \(y,z\in I\) imply \(xy,y(zx)\in I\). \(B\subseteq A\) is called a congruence kernel of \(\mathcal A\) if there exists a congruence \(\Theta \) on \(\mathcal A\) with \([1]\Theta =B\). It is proved that for pre-logics the notions of deductive system, ideal and congruence kernel coincide. It is proved that the lattice of all deductive systems of \(\mathcal A\) is distributive and algebraic and hence relatively pseudocomplemented. Principal deductive systems are described and it is shown that to each quasiordered set a pre-logic can be assigned in a natural way.

MSC:
08A30 Subalgebras, congruence relations
03G25 Other algebras related to logic
03B20 Subsystems of classical logic (including intuitionistic logic)
03B22 Abstract deductive systems
06D15 Pseudocomplemented lattices
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