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Discussiones Mathematicae General Algebra and Applications 22(2) (2002) 183-198
DOI: https://doi.org/10.7151/dmgaa.1057

QUASI-IMPLICATION ALGEBRAS

Ivan Chajda and Kamil Dusek

Department of Algebra and Geometry
Palacký University of Olomouc
Tomkova 40, CZ-77900 Olomouc, Czech Republic
e-mail: chajda@risc.upol.cz

Abstract

A quasi-implication algebra is introduced as an algebraic counterpart of an implication reduct of propositional logic having non-involutory negation (e.g. intuitionistic logic). We show that every pseudocomplemented semilattice induces a quasi-implication algebra (but not conversely). On the other hand, a more general algebra, a so-called pseudocomplemented q-semilattice is introduced and a mutual correspondence between this algebra and a quasi-implication algebra is shown.

Keywords: implication, non-involutory negation, quasi-implication algebra, implitcation algebra, pseudocomplemented semilattice, q-semilattice.

2000 Mathematics Subject Classification: 03G25, 06A12, 06D15.

References

[1] J.C. Abbott, Semi-boolean algebras , Mat. Vesnik 4 (1967), 177-198.

[2] R. Balbes, On free pseudo-complemented and relatively pseudo-complemented semi-lattices , Fund. Math. 78 (1973), 119-131.

[3] I. Chajda, Semi-implication algebra , Tatra Mt. Math. Publ. 5 (1995), 13-24.

[4] I. Chajda, An extension of relative pseudocomplementation to non-distributive lattices , Acta Sci. Math. (Szeged), to appear.

[5] A. Diego, Sur les algèbres de Hilbert , Gauthier-Villars, Paris 1966 (viii+55pp.).

[6] O. Frink, Pseudo-complements in semi-lattices , Duke Math. J. 29 (1962), 505-514.

Received 10 October 2002
Revised 23 January 2003