Varieties of De Morgan monoids : covers of atoms

dc.contributor.authorMoraschini, Tommaso
dc.contributor.authorRaftery, James G.
dc.contributor.authorWannenburg, Johann Joubert
dc.contributor.emailjames.raftery@up.ac.zaen_ZA
dc.date.accessioned2021-04-14T11:45:44Z
dc.date.issued2020
dc.description.abstractThe variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4{element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0{generated algebra onto which nitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety V(C4) within U are revealed here. There are just ten of them (all nitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety|in fact, every nite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM are discriminator varieties. Of these, we identify in nitely many that are nitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.en_ZA
dc.description.departmentMathematics and Applied Mathematicsen_ZA
dc.description.embargo2021-06-01
dc.description.librarianam2021en_ZA
dc.description.sponsorshipThe European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.en_ZA
dc.description.urihttps://www.cambridge.org/core/journals/review-of-symbolic-logicen_ZA
dc.identifier.citationMoraschini, T., Raftery, J.G. & Wannenburg, J.J. 2020, 'Varieties of De Morgan Monoids : covers of atoms', Review of Symbolic Logic, vol. 13, no. 2, pp. 16349-16360.en_ZA
dc.identifier.issn1755-0203 (print)
dc.identifier.issn1755-0211 (online)
dc.identifier.other10.1017/S1755020318000448
dc.identifier.urihttp://hdl.handle.net/2263/79444
dc.language.isoenen_ZA
dc.publisherCambridge University Pressen_ZA
dc.rights© Association for Symbolic Logic 2020en_ZA
dc.subjectDe Morgan monoiden_ZA
dc.subjectSugihara monoiden_ZA
dc.subjectDunn monoiden_ZA
dc.subjectResiduated latticeen_ZA
dc.subjectRelevance logicen_ZA
dc.titleVarieties of De Morgan monoids : covers of atomsen_ZA
dc.typeArticleen_ZA

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