Residuated lattices derived from filters(ideals) in double Boolean algebras

Document Type : Original Article


1 University of Yaound´e 1, Faculty of Sciences, Department of Mathematics, Laboratory of Algebra, P.O. Box 812, Yaound´e, Republic of Cameroon

2 Ecole Normale Sup´erieure de Yaound´e, Department of Mathematics, Cameroon


Double Boolean algebras (dBas) are algebraic structures D = (D, v, ^, ‌, ', ⊥, T) of type (2, 2, 1, 1, 0, 0), introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Our goal is an algebraic investigation of  dBas, based on similar results on Boolean algebras. In this paper, first we characterize filters on dBas as deductive systems and we give many characterization of primary filters(ideals). Second, for a given dBa, we show that the set of its filters F(D) (resp.ideals I(D)) is endowed with the structure of distributive pseudo-complemented lattices, Heyting algebras and residuated lattices. We finish by introducing the notions of annihilators and co-annihilators on dBas and investigate some relalted properties of them. We show that pseudo-complement of an ideal I (filter F) is the annihilator I* of I ( co-annihilator F*) and the set of annihilators (co-annihilators) forms a Boolean algebra. 


[1] R. Beazer, Congruence pairs of distributive double p-algebras with non-empty core, Houston
Journal of Mathematics, 6(4) (1980), 1–12.
[2] B.E. Breckner, S. Christian, Topological representation of double Boolean lattices, Studia
Universitatis Babes̆-Bolyai Mathematica, 64(1) (2019), 11–23.
[3] S. Burris, H.P. Sankappanavar, A course in universal algebra, Springer-Verlag, New York,
[4] W.H. Cornish, Annulets and α-ideals in a distributive lattice, Journal of the Australian Mathematical
Society, 15(1) (1973), 70–77.
[5] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Studies in logic and the foundations of mathematics,
Elsevier, 151 (2007), 1–509.
[6] B. Ganter, R. Wille, Formal concept analysis, Mathematical Foundations, Springer, Heidelberg,
[7] G.Grätzer, A general lattice theory, Academic Press, New York, San Francisco, 1987.
[8] G. Grätzer, Lattice theory: Foundation, Springer Basel AG, 2011.
[9] P. Howlader, M. Banerjee, Remarks on prime ideal and representation theorem for double
Boolean algebras, The 15th International Conference on Concept lattices and their Applications,
(CLA 2020), (2020), 83–94.
[10] P. Howlader, M. Banerjee, Topological representation of double Boolean algebras, ArXiv:2103.
[11] M. Kondo, E. Turunen, Prime filters on residuated lattices, Proceedings of International
Simposium on Multiple-Valued Logic, 2012.
[12] L. Kwuida, Prime ideal theorem for double Boolean algebras, Discussiones Mathematicae-
General Algebra and Applications, 27(2) (2005), 263–275.
[13] M. Mandelker, Relative annihilators in lattices, Duke Mathematcal Journal, 37(2) (1970),
[14] Y.L.J. Tenkeu, E.R.A. Temgoua, L. Kwuida, Filters, ideals and congruences on double Boolean
algebras, Springer Nature Switzerland AG 2021, A. Braud et al.(Eds.): ICFCA 2021, LNAI
12733, (2021), 270–280.
[15] Y.L.J. Tenkeu, E.R.A. Temgoua, L. Kwuida, Filters, ideals and powers of double Boolean
algebras, Submitted.
[16] R. Wille, Boolean concept logic, In B. Ganter and G.W. Mineau, Conceptual Structures:
Logical Linguistic, and Computational Issue, Springer, Berlin Heidelberg, (2000), 317–331.