The Reticulation of a Universal Algebra

Published in Volume XXVIII, Issue 1, 2018, pages 67–113, doi: 10.7561/SACS.2018.1.67

Authors: G. Georgescu, C. Mureșan

Abstract

The reticulation of an algebra A is a bounded distributive lattice L(A) whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of A, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra A from a semi–degenerate congruence–modular variety C in the case when the commutator of A, applied to compact congruences of A, produces compact congruences, in particular when C has principal commutators; furthermore, it turns out that weaker conditions than the above are sufficient for A to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL–algebra and that of an MV–algebra. The purpose of constructing the reticulation for the algebras from C is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and C, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.

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Bibtex

@article{sacscuza:georgescu2018troaua,
  title={The Reticulation of a Universal Algebra},
  author={G. Georgescu and C. Mure{c s}an},
  journal={Scientific Annals of Computer Science},
  volume={28},
  number={1},
  organization={``A.I. Cuza'' University, Iasi, Romania},
  year={2018},
  pages={67–113},
  doi={10.7561/SACS.2018.1.67},
  publisher={``A.I. Cuza'' University Press}
}