1. bookVolume 42 (2022): Issue 1 (March 2022)
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
access type Open Access

Stone Commutator Lattices and Baer Rings

Published Online: 05 Apr 2022
Volume & Issue: Volume 42 (2022) - Issue 1 (March 2022)
Page range: 51 - 96
Received: 08 Apr 2019
Accepted: 30 Nov 2020
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
Abstract

In this paper, we transfer Davey‘s characterization for κ -Stone bounded distributive lattices to lattices with certain kinds of quotients, in particular to commutator lattices with certain properties, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of these lattices. We then apply these results to certain congruence lattices, in particular to those of semiprime members of semi-degenerate congruence-modular varieties, and use this particular case to transfer Davey‘s Theorem to commutative unitary rings.

Keywords

MSC 2010

[1] P. Agliano, Prime spectra in modular varieties, Algebra Univ. 30 (1993) 581–597. https://doi.org/10.1007/BF0119538310.1007/BF01195383 Search in Google Scholar

[2] E. Aichinger, Congruence lattices forcing nilpotency, J. Algebra and Its Appl. 17 (2) (2018). https://doi.org/10.1142/S021949881850033010.1142/S0219498818500330 Search in Google Scholar

[3] L.P. Belluce, Spectral spaces and non-commutative rings, Commun. in Algebra 19 (7) (1991) 1855–1865. https://doi.org/10.1080/0092787910882423410.1080/00927879108824234 Search in Google Scholar

[4] L.P. Belluce, Spectral closure for non-commutative rings, Commun. in Algebra 25 (5) (1997) 1513–1536. https://doi.org/10.1080/0092787970882593310.1080/00927879708825933 Search in Google Scholar

[5] J. Czelakowski, Additivity of the commutator and residuation, Rep. Math. Logic 43 (2008) 109–132. https://doi.org/10.1007/978-3-319-21200-510.1007/978-3-319-21200-5 Search in Google Scholar

[6] B.A. Davey, m-stone lattices, Canad. J. Math. 24 (6) (1972) 1027–1032. https://doi.org/10.4153/CJM-1972-104-x10.4153/CJM-1972-104-x Search in Google Scholar

[7] R. Freese and R. McKenzie, Commutator Theory for Congruence-modular Varieties, London Mathematical Society Lecture Note Series 125 (Cambridge University Press, 1987). https://doi.org/10.1112/blms/20.4.36210.1112/blms/20.4.362 Search in Google Scholar

[8] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic and The Foundations of Mathematics 151 (Elsevier, Amsterdam/Boston /Heidelberg /London /New York /Oxford /Paris /San Diego/San Francisco /Singapore /Sydney /Tokyo, 2007). https://doi.org/10.1016/s0049-237x(07)x8002-910.1016/S0049-237X(07)X8002-9 Search in Google Scholar

[9] G. Georgescu and C. Mureşan, Congruence Boolean lifting property, J. Multiple-Valued Logic and Soft Comp. 29 (3–4) (2017) 225–274. Search in Google Scholar

[10] G. Georgescu and C. Mureşan, The reticulation of a universal algebra, Sci. Ann. Comp. Sci. XXVIII (2018) 67–113. https://doi.org/10.7561/SACS.2018.1.6710.7561/SACS.2018.1.67 Search in Google Scholar

[11] J.E. Kist, Two characterizations of commutative Baer rings, Pacific J. Math. 50 (1974). https://doi.org/10.2140/pjm.1974.50.12510.2140/pjm.1974.50.125 Search in Google Scholar

[12] A. Iorgulescu, Algebras of Logic as BCK Algebras (Editura ASE, Bucharest, 2008). Search in Google Scholar

[13] A. Joyal, Le théorème de Chevalley-Tarski et Remarques sur l‘algèbre constructive, Cahiers Topol. Géom. Différ. 16 (1975) 256–258. Search in Google Scholar

[14] C. Mureşan, Algebras of Many-valued Logic. Contributions to the Theory of Residuated Lattices, Ph.D. Thesis, 2009. Search in Google Scholar

[15] C. Mureşan, Co-stone residuated lattices, Annals of the University of Craiova, Math. Comp. Sci. Ser. 40 (2013) 52–75. https://doi.org/10.7561/SACS.2018.1.6710.7561/SACS.2018.1.67 Search in Google Scholar

[16] P. Ouwehand, Commutator Theory and Abelian Algebras. arXiv:1309.0662 [math.RA]. Search in Google Scholar

[17] D. Piciu, Algebras of Fuzzy Logic (Editura Universitaria Craiova, Craiova, 2007). Search in Google Scholar

[18] D. Schweigert, Tolerances and commutators on lattices, Bull. Austral. Math. Soc. 37 (2) (1988) 213–219. https://doi.org/10.1017/S000497270002674510.1017/S0004972700026745 Search in Google Scholar

[19] H. Simmons, Reticulated rings, J. Algebra 66 (1980) 169–192. https://doi.org/10.1016/0021-8693(80)90118-010.1016/0021-8693(80)90118-0 Search in Google Scholar

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