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.
@article{bwmeta1.element.doi-10_2478_s11533-009-0002-y,
author = {Abad Manuel and Cimadamore Cecilia and D\'\i az Varela Jos\'e},
title = {Topological representation for monadic implication algebras},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {299-309},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0002-y}
}
Abad Manuel; Cimadamore Cecilia; Díaz Varela José. Topological representation for monadic implication algebras. Open Mathematics, Tome 7 (2009) pp. 299-309. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0002-y/
[00000] [1] Abad M., Díaz Varela J.P., Zander M., Varieties and quasivarieties of monadic Tarski algebras, Sci. Math. Jpn., 2002, 56(3), 599–612
[00001] [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]
[00002] [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]
[00003] [4] Abbott J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1967, 11(59), 3–23
[00004] [5] Burris S., Sankappanavar H.P., A course in universal algebra, Graduate Texts in Mathematics 78, Springer-Verlag, New York-Berlin, 1981
[00005] [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]
[00006] [7] Halmos P.R., Algebraic logic I, Monadic Boolean algebras, Compositio Math., 1956, 12, 217–249
[00007] [8] Halmos P.R., Free monadic algebras, Proc. Amer. Math. Soc., 1959, 10, 219–227 http://dx.doi.org/10.2307/2033581[Crossref]
[00008] [9] Halmos P.R., Algebraic logic, Chelsea Publishing Co., New York, 1962
[00009] [10] Iturrioz L., Monteiro A., Représentation des algèbres de Tarski monadiques, preprint
[00010] [11] Koppelberg S., Handbook of Boolean algebras, North-Holland Publishing Co., Amsterdam, 1989