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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 $$\Theta =B$$. 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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##### References:
  ABOTT J. C.: Semi-boolean algebras. Mat. Vesnik 4 (1967), 177-198.  CHAJDA I.: Semi-implication algebra. Tatra Mt. Math. Publ. 5 (1995), 13-24. · Zbl 0856.08004  CHAJDA I.: The lattice of deductive systems on Hilbert algebras. Southeast Asian Bull. Math. · Zbl 1010.03054  CHAJDA I., HALAŠ R.: Congruences and ideals in Hilbert algebras. Kyungpook Math. J. 39 (1999), 429-432. · Zbl 0954.08002  CHAJDA I., HALAŠ R.: Stabilizers in Hilbert algebras. Multiple Valued Logic · Zbl 1024.03065  CHAJDA I., HALAŠ R.: Order algebras. Demonstrate Math. 35 (2002), 1-10. · Zbl 1236.08004  CHAJDA I., HALAŠ R., ZEDNÍK J.: Filters and annihilators in implication algebras. Acta Univ. Palack. Olomouc. Fac. Rerum Natur. Math. 37 (1998), 41-45. · Zbl 0967.03059  DIEGO A.: Sur les algébres de Hilbert. Collection de Logique Math. Ser. A. 21, Hermann, Paris, 1967, pp. 177-198.  DUDEK W. A.: On ideals in Hilbert algebras. Acta Univ. Palack. Olomouc. Fac. Rerum Natur. Math. 38 (1999), 31-34. · Zbl 0957.06018  JUN Y. B.: Deductive systems of Hilbert algebras. Math. Japon. 43 (1996), 51-54. · Zbl 0946.03079
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