Mathematicae General Algebra and Applications 22(2) (2002)
Ivan Chajda and Kamil Dusek
Department of Algebra and Geometry
Palacký University of Olomouc
Tomkova 40, CZ-77900 Olomouc, Czech Republic
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.
J.C. Abbott, Semi-boolean algebras , Mat. Vesnik 4 (1967), 177-198.
R. Balbes, On free pseudo-complemented and relatively pseudo-complemented semi-lattices , Fund.
Math. 78 (1973), 119-131.
I. Chajda, Semi-implication algebra , Tatra Mt. Math. Publ. 5 (1995), 13-24.
I. Chajda, An extension of relative pseudocomplementation to non-distributive lattices , Acta Sci.
Math. (Szeged), to appear.
A. Diego, Sur les algèbres de Hilbert , Gauthier-Villars, Paris 1966 (viii+55pp.).
O. Frink, Pseudo-complements in semi-lattices , Duke Math. J. 29 (1962), 505-514.
Received 10 October 2002
Revised 23 January 2003