Quasi-implication algebras

Ivan Chajda; Kamil Dušek

Discussiones Mathematicae - General Algebra and Applications (2002)

  • Volume: 22, Issue: 2, page 183-198
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Ivan Chajda, and Kamil Dušek. "Quasi-implication algebras." Discussiones Mathematicae - General Algebra and Applications 22.2 (2002): 183-198. <http://eudml.org/doc/287650>.

@article{IvanChajda2002,
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.},
author = {Ivan Chajda, Kamil Dušek},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {implication; non-involutory negation; quasi-implication algebra; implitcation algebra; pseudocomplemented semilattice; q-semilattice},
language = {eng},
number = {2},
pages = {183-198},
title = {Quasi-implication algebras},
url = {http://eudml.org/doc/287650},
volume = {22},
year = {2002},
}

TY - JOUR
AU - Ivan Chajda
AU - Kamil Dušek
TI - Quasi-implication algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2002
VL - 22
IS - 2
SP - 183
EP - 198
AB - 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.
LA - eng
KW - implication; non-involutory negation; quasi-implication algebra; implitcation algebra; pseudocomplemented semilattice; q-semilattice
UR - http://eudml.org/doc/287650
ER -

References

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  1. [1] J.C. Abbott, Semi-boolean algebras, Mat. Vesnik 4 (1967), 177-198. Zbl0153.02704
  2. [2] R. Balbes, On free pseudo-complemented and relatively pseudo-complemented semi-lattices, Fund. Math. 78 (1973), 119-131. Zbl0277.06001
  3. [3] I. Chajda, Semi-implication algebra, Tatra Mt. Math. Publ. 5 (1995), 13-24. Zbl0856.08004
  4. [4] I. Chajda, An extension of relative pseudocomplementation to non-distributive lattices, Acta Sci. Math. (Szeged), to appear. Zbl1048.06005
  5. [5] A. Diego, Sur les algèbres de Hilbert, Gauthier-Villars, Paris 1966, (viii+55pp.). 
  6. [6] O. Frink, Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505-514. Zbl0114.01602

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