The pure spectrum of a residuated lattice
Abstract
This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination of algebraic and topological methods on the pure filters of a residuated lattice is applied to obtain some new structural results. The notion of purely-prime filters of a residuated lattice has been investigated, and a Cohen-type theorem has been obtained. It is shown that the pure spectrum of a residuated lattice is a compact sober space, and a Grothendieck-type theorem has been demonstrated. It is proved that the pure spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is investigated and verified that a given residuated lattice is mp iff its minimal prime spectrum is equipped with the induced dual hull-kernel topology, and its pure spectrum is the same.
Keywords
Cite
@article{arxiv.2207.07995,
title = {The pure spectrum of a residuated lattice},
author = {Saeed Rasouli and Amin Dehghani},
journal= {arXiv preprint arXiv:2207.07995},
year = {2022}
}
Comments
43 pages, 5 figures, original paper. arXiv admin note: substantial text overlap with arXiv:2202.10117, arXiv:2203.15018