Semidegenerate Congruence-modular Algebras Admitting a Reticulation
The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spe...
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| Format: | Article |
| Language: | English |
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Alexandru Ioan Cuza University of Iasi
2023-05-01
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| Series: | Scientific Annals of Computer Science |
| Online Access: | https://www.info.uaic.ro/en/sacs_articles/semidegenerate-congruence-modular-algebras-admitting-a-reticulation/ |
| _version_ | 1797816691310198784 |
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| author | George Georgescu |
| author_facet | George Georgescu |
| author_sort | George Georgescu |
| collection | DOAJ |
| description | The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construction of the lattice L(R) was generalized by Belluce for each unital ring R and the reticulation was defined by axioms. In a recent paper we generalized the Belluce construction for algebras in a semidegenerate congruence-modular variety V. For any algebra A ∈ V we defined a bounded distributive lattice L(A), but in general the prime spectrum Spec(A) of A is not homeomorphic with the prime spectrum SpecId (L(A)). We introduced the quasi-commutative algebras in the variety V (as a generalization of Belluce’s quasi-commutative rings) and proved that for any algebra A ∈ V, the spectra Spec(A) and SpecId (L(A)) are homeomorphic. In this paper we define the reticulation A ∈ V by four axioms and prove that any two reticulations of A are isomorphic lattices. By using the uniqueness of reticulation and other results from the mentioned paper, we obtain a characterization theorem for the algebras A ∈ V that admit a reticulation: A is quasi-commutative if and only if A admits a reticulation. This result is a universal algebra generalization of the following Belluce theorem: a ring R is quasi-commutative if and only if R admits a reticulation. Another subject treated in this paper is the spectral closure of the prime spectrum Spec(A) of an algebra A ∈ V, a notion that generalizes the Belluce spectral closure of the prime spectrum of a ring. |
| first_indexed | 2024-03-13T08:41:27Z |
| format | Article |
| id | doaj.art-ca77d13654444e37addd9e0179840c6f |
| institution | Directory Open Access Journal |
| issn | 1843-8121 2248-2695 |
| language | English |
| last_indexed | 2024-03-13T08:41:27Z |
| publishDate | 2023-05-01 |
| publisher | Alexandru Ioan Cuza University of Iasi |
| record_format | Article |
| series | Scientific Annals of Computer Science |
| spelling | doaj.art-ca77d13654444e37addd9e0179840c6f2023-05-30T11:14:33ZengAlexandru Ioan Cuza University of IasiScientific Annals of Computer Science1843-81212248-26952023-05-01XXXIII153410.7561/SACS.2023.1.5Semidegenerate Congruence-modular Algebras Admitting a ReticulationGeorge Georgescuhttps://orcid.org/0000-0003-2294-6289 The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construction of the lattice L(R) was generalized by Belluce for each unital ring R and the reticulation was defined by axioms. In a recent paper we generalized the Belluce construction for algebras in a semidegenerate congruence-modular variety V. For any algebra A ∈ V we defined a bounded distributive lattice L(A), but in general the prime spectrum Spec(A) of A is not homeomorphic with the prime spectrum SpecId (L(A)). We introduced the quasi-commutative algebras in the variety V (as a generalization of Belluce’s quasi-commutative rings) and proved that for any algebra A ∈ V, the spectra Spec(A) and SpecId (L(A)) are homeomorphic. In this paper we define the reticulation A ∈ V by four axioms and prove that any two reticulations of A are isomorphic lattices. By using the uniqueness of reticulation and other results from the mentioned paper, we obtain a characterization theorem for the algebras A ∈ V that admit a reticulation: A is quasi-commutative if and only if A admits a reticulation. This result is a universal algebra generalization of the following Belluce theorem: a ring R is quasi-commutative if and only if R admits a reticulation. Another subject treated in this paper is the spectral closure of the prime spectrum Spec(A) of an algebra A ∈ V, a notion that generalizes the Belluce spectral closure of the prime spectrum of a ring.https://www.info.uaic.ro/en/sacs_articles/semidegenerate-congruence-modular-algebras-admitting-a-reticulation/ |
| spellingShingle | George Georgescu Semidegenerate Congruence-modular Algebras Admitting a Reticulation Scientific Annals of Computer Science |
| title | Semidegenerate Congruence-modular Algebras Admitting a Reticulation |
| title_full | Semidegenerate Congruence-modular Algebras Admitting a Reticulation |
| title_fullStr | Semidegenerate Congruence-modular Algebras Admitting a Reticulation |
| title_full_unstemmed | Semidegenerate Congruence-modular Algebras Admitting a Reticulation |
| title_short | Semidegenerate Congruence-modular Algebras Admitting a Reticulation |
| title_sort | semidegenerate congruence modular algebras admitting a reticulation |
| url | https://www.info.uaic.ro/en/sacs_articles/semidegenerate-congruence-modular-algebras-admitting-a-reticulation/ |
| work_keys_str_mv | AT georgegeorgescu semidegeneratecongruencemodularalgebrasadmittingareticulation |