Discriminator varieties of Boolean algebras with residuated operators

Peter Jipsen

Banach Center Publications (1993)

  • Volume: 28, Issue: 1, page 239-252
  • ISSN: 0137-6934

Abstract

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The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jónsson.

How to cite

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Jipsen, Peter. "Discriminator varieties of Boolean algebras with residuated operators." Banach Center Publications 28.1 (1993): 239-252. <http://eudml.org/doc/262745>.

@article{Jipsen1993,
abstract = {The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jónsson.},
author = {Jipsen, Peter},
journal = {Banach Center Publications},
keywords = {Boolean algebras with a residuated binary operator; commutative residuated monoids; discriminator algebras; equational bases; varieties of -algebras; residuated Boolean monoids},
language = {eng},
number = {1},
pages = {239-252},
title = {Discriminator varieties of Boolean algebras with residuated operators},
url = {http://eudml.org/doc/262745},
volume = {28},
year = {1993},
}

TY - JOUR
AU - Jipsen, Peter
TI - Discriminator varieties of Boolean algebras with residuated operators
JO - Banach Center Publications
PY - 1993
VL - 28
IS - 1
SP - 239
EP - 252
AB - The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jónsson.
LA - eng
KW - Boolean algebras with a residuated binary operator; commutative residuated monoids; discriminator algebras; equational bases; varieties of -algebras; residuated Boolean monoids
UR - http://eudml.org/doc/262745
ER -

References

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  9. [M75] R. McKenzie, On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model, J. Symbolic Logic 40 (2) (1975), 186-196. Zbl0316.02052
  10. [MMT] R. N. McKenzie, G. F. McNulty and W. F. Taylor, Algebras, Lattices, Varieties, Volume I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth, Belmont, California, 1987. 
  11. [P61] W. Prenowitz, A contemporary approach to classical geometry, Amer. Math. Monthly 68 (No. 1, Part II) (1961). Zbl0094.15402
  12. [W78] H. Werner, Discriminator algebras, Stud. Algebra Anwendungen 6, Akademie-Verlag, Berlin 1978. 

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