New topology in residuated lattices

Abstract

In this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$. Finally, we study the uniform topology $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices

“…For examples of residuated lattices see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. We denote residuated lattices by RL.…”

“…We note that in De Morgan residuated lattices the notions of ideals and filters are not dual. In 2018, Holdon [11] constructed a new topology based on upsets (filters) in residuated lattices, in a dual manner, using downsets (ideals), we can study a dual topology.…”