# The Belluce lattice associated with a bounded BCK -algebra

Authors

Faculty of Sciences, Department of Mathematics, University of Craiova, Craiova, Romania

Abstract

In this paper, we introduce the notions of Belluce lattice associated with a bounded $BCK$-algebra and reticulation of a bounded $BCK$-algebra. To do this, first, we define the operations  $\curlywedge ,$ $\curlyvee$ and $\sqcup$ on $BCK$-algebras and we study some algebraic properties of them. Also, for a bounded $BCK$-algebra $A$ we define the Zariski topology on $\ Spec(A)$ and the induced topology $\tau _{A,Max(A)}$ on $Max(A)$. We prove $(Max(A),\tau_{A,Max(A)})$ is a compact topological space if $A$ has Glivenko property. Using the open and the closed sets of $Max(A)$, we define a congruence relation on a bounded $BCK$-algebra $A$ and we show $L_{A}$, the quotient set, is a bounded distributive lattice. We call this lattice the Belluce lattice associated with $A.$ Finally, we show $(L_{A},p_{A})$ is a reticulation of $A$ (in the sense of Definition \ref{d7}) and the lattices $L_{A}$ and $S_{A}$ are isomorphic.

Keywords

#### References

[1] M.F. Atiyah, I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Series in
[2] R. Balbes, P. Dwinger, Distributive lattices, University of Missouri Press, XIII, 1974.
[3] L.P. Belluce, Semisimple algebras of in nite-valued logic and bold fuzzy set theory, Canadian
Journal of Mathematics, 38(6) (1986), 1356{1379.
[4] L.P. Belluce, Spectral spaces and non-commutative rings, Communications in Algebra, 19
(1991), 1855{1865.
[5] D. Busneag, D. Piciu, The Belluce lattice associated with a bounded Hilbert algebra, Soft
Computing, 19 (2015), 3031{3042.
[6] S.A. Celani, Deductive systems of BCK-algebras, Acta Univ. Palackianae Olomoucensis, Facultas
Rervoum Naturalium Mathematisa, 43(1) (2004), 27{32.
[7] L. Ciungu, Non-commutative multiple-valued logic algebras, Springer International Publishing,
2014.
[8] G. Georgescu, The reticulation of a quantale, Revue Roumaine des Mathematiques Pures et
Appliquees, 7-8 (1995), 619{631.
[9] M. Ghita, Contributions to the study of the category of Hilbert algebras, Ph. D. Thesis, University
of Craiova, 2011.
[10] K. Iseki, On some ideals in BCK-algebras, Mathematics Seminar Notes, 3 (1975), 65{70.
[11] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica,
23(1) (1978), 1{26.
[12] J. Kuhr, Pseudo BCK-algebras and related structures, Univerzita Palackeho v Olomouci,
2007.
[13] L. Leustean, The prime and maximal spectra and the reticulation of BL-algebras, Central
European Journal of Mathematics, 3 (2003), 382{397.
[14] C. Muresan, The reticulation of a residuated lattice, Bulletin Mathematique de la Societe des
Sciences Mathematiques de Roumanie, 51(1) (2008), 47{65.
[15] C. Muresan, Characterization of the reticulation of a residuated lattice, Journal of Multiple-
Valued Logic and Soft Computing, 16 (2010), 427{447.
[16] A.B. Saeid, C. Flaut, S. Hoskova-Mayerova, M. Afshar, Some connections between BCK-
algebras and n-ary block codes, Soft Computing, 22(1) (2018), 41{46.
[17] H. Simmons, Reticulated rings, Journal of Algebra, 66 (1980), 169{192.