Topological representation for monadic implication algebras

Abad Manuel; Cimadamore Cecilia; Díaz Varela José

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 299-309
  • ISSN: 2391-5455

Abstract

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In this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.

How to cite

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Abad Manuel, Cimadamore Cecilia, and Díaz Varela José. "Topological representation for monadic implication algebras." Open Mathematics 7.2 (2009): 299-309. <http://eudml.org/doc/269669>.

@article{AbadManuel2009,
abstract = {In this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.},
author = {Abad Manuel, Cimadamore Cecilia, Díaz Varela José},
journal = {Open Mathematics},
keywords = {Implication algebra; Monadic Boolean algebra; Implication spaces; Dual categorical equivalence; implication algebra; monadic implication algebra; topology; implication space; categorical equivalence},
language = {eng},
number = {2},
pages = {299-309},
title = {Topological representation for monadic implication algebras},
url = {http://eudml.org/doc/269669},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Abad Manuel
AU - Cimadamore Cecilia
AU - Díaz Varela José
TI - Topological representation for monadic implication algebras
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 299
EP - 309
AB - In this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.
LA - eng
KW - Implication algebra; Monadic Boolean algebra; Implication spaces; Dual categorical equivalence; implication algebra; monadic implication algebra; topology; implication space; categorical equivalence
UR - http://eudml.org/doc/269669
ER -

References

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  1. [1] Abad M., Díaz Varela J.P., Zander M., Varieties and quasivarieties of monadic Tarski algebras, Sci. Math. Jpn., 2002, 56(3), 599–612  
  2. [2] Abad M., Díaz Varela J.P., Torrens A., Topological representation for implication algebras, Algebra Universalis, 2004, 52, 39–48 http://dx.doi.org/10.1007/s00012-004-1872-2[Crossref] 
  3. [3] Abad M., Monteiro L., Savini S., Seewald J., Free monadic Tarski algebras, Algebra Universalis, 1997, 37, 106–118 http://dx.doi.org/10.1007/s000120050006[Crossref] 
  4. [4] Abbott J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1967, 11(59), 3–23  
  5. [5] Burris S., Sankappanavar H.P., A course in universal algebra, Graduate Texts in Mathematics 78, Springer-Verlag, New York-Berlin, 1981  
  6. [6] Cignoli R., Quantifiers on distributive lattices, Discrete Mathematics, 1991, 96, 183–197 http://dx.doi.org/10.1016/0012-365X(91)90312-P[Crossref] 
  7. [7] Halmos P.R., Algebraic logic I, Monadic Boolean algebras, Compositio Math., 1956, 12, 217–249  
  8. [8] Halmos P.R., Free monadic algebras, Proc. Amer. Math. Soc., 1959, 10, 219–227 http://dx.doi.org/10.2307/2033581[Crossref] 
  9. [9] Halmos P.R., Algebraic logic, Chelsea Publishing Co., New York, 1962  
  10. [10] Iturrioz L., Monteiro A., Représentation des algèbres de Tarski monadiques, preprint  
  11. [11] Koppelberg S., Handbook of Boolean algebras, North-Holland Publishing Co., Amsterdam, 1989  

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