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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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