The Reticulation of a Universal Algebra
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...
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Alexandru Ioan Cuza University of Iasi
2018-06-01
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Series: | Scientific Annals of Computer Science |
Online Access: | http://www.info.uaic.ro/bin/download/Annals/XXVIII1/XXVIII1_2.pdf |
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doaj-4d9fec4ca91d464faf3b368cdd4300b22020-11-25T01:06:44ZengAlexandru Ioan Cuza University of IasiScientific Annals of Computer Science1843-81212248-26952018-06-01XXVIII16711310.7561/SACS.2018.1.67The Reticulation of a Universal AlgebraG. GeorgescuC. MureșanThe 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.http://www.info.uaic.ro/bin/download/Annals/XXVIII1/XXVIII1_2.pdf |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
G. Georgescu C. Mureșan |
spellingShingle |
G. Georgescu C. Mureșan The Reticulation of a Universal Algebra Scientific Annals of Computer Science |
author_facet |
G. Georgescu C. Mureșan |
author_sort |
G. Georgescu |
title |
The Reticulation of a Universal Algebra |
title_short |
The Reticulation of a Universal Algebra |
title_full |
The Reticulation of a Universal Algebra |
title_fullStr |
The Reticulation of a Universal Algebra |
title_full_unstemmed |
The Reticulation of a Universal Algebra |
title_sort |
reticulation of a universal algebra |
publisher |
Alexandru Ioan Cuza University of Iasi |
series |
Scientific Annals of Computer Science |
issn |
1843-8121 2248-2695 |
publishDate |
2018-06-01 |
description |
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. |
url |
http://www.info.uaic.ro/bin/download/Annals/XXVIII1/XXVIII1_2.pdf |
work_keys_str_mv |
AT ggeorgescu thereticulationofauniversalalgebra AT cmuresan thereticulationofauniversalalgebra AT ggeorgescu reticulationofauniversalalgebra AT cmuresan reticulationofauniversalalgebra |
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