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Varieties of De Morgan Monoids
Thesis (PhD)--University of Pretoria, 2020.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2020
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| _version_ | 1867613713921474560 |
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| access_status_str | Open Access |
| author2 | Raftery, James G. |
| author_browse | Raftery, James G. |
| author_facet | Raftery, James G. |
| collection | Thesis |
| dc_rights_str_mv | © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Thesis (PhD)--University of Pretoria, 2020. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/75178 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:40:31.851Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/75178 Varieties of De Morgan Monoids Raftery, James G. jamiewannenburg@gmail.com Moraschini, Tommaso Wannenburg, Johann Joubert Abstract algebraic logic De Morgan monoids Relevance logic Universal algebra Beth property UCTD Thesis (PhD)--University of Pretoria, 2020. De Morgan monoids are algebraic structures that model certain non-classical logics. The variety DMM of all De Morgan monoids models the relevance logic Rt (so-named because it blocks the derivation of true conclusions from irrelevant premises). The so-called subvarieties and subquasivarieties of DMM model the strengthenings of Rt by new logical axioms, or new inference rules, respectively. Meta-logical problems concerning these stronger systems amount to structural problems about (classes of) De Morgan monoids, and the methods of universal algebra can be exploited to solve them. Until now, this strategy was under-developed in the case of Rt and DMM. The thesis contributes in several ways to the filling of this gap. First, a new structure theorem for irreducible De Morgan monoids is proved; it leads to representation theorems for the algebras in several interesting subvarieties of DMM. These in turn help us to analyse the lower part of the lattice of all subvarieties of DMM. This lattice has four atoms, i.e., DMM has just four minimal subvarieties. We describe in detail the second layer of this lattice, i.e., the covers of the four atoms. Within certain subvarieties of DMM, our description amounts to an explicit list of all the covers. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. Thereafter, we use these insights to identify strengthenings of Rt with certain desirable meta-logical features. In each case, we work with the algebraic counterpart of a meta-logical property. For example, we identify precisely the varieties of De Morgan monoids having the joint embedding property (any two nontrivial members both embed into some third member), and we establish convenient sufficient conditions for epimorphisms to be surjective in a subvariety of DMM. The joint embedding property means that the corresponding logic is determined by a single set of truth tables. Epimorphisms are related to 'implicit definitions'. (For instance, in a ring, the multiplicative inverse of an element is implicitly defined, because it is either uniquely determined or non-existent.) The logical meaning of epimorphism-surjectivity is, roughly speaking, that suitable implicit definitions can be made explicit in the corresponding logical syntax. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Mathematics and Applied Mathematics PhD Unrestricted 2020-07-13T13:55:46Z 2020-07-13T13:55:46Z 2020-09-01 2020 Thesis Wannenburg, JJ 2020, Varieties of De Morgan Monoids, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/75178> S2020 http://hdl.handle.net/2263/75178 en © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | Abstract algebraic logic De Morgan monoids Relevance logic Universal algebra Beth property UCTD Varieties of De Morgan Monoids |
| title | Varieties of De Morgan Monoids |
| title_full | Varieties of De Morgan Monoids |
| title_fullStr | Varieties of De Morgan Monoids |
| title_full_unstemmed | Varieties of De Morgan Monoids |
| title_short | Varieties of De Morgan Monoids |
| title_sort | varieties of de morgan monoids |
| topic | Abstract algebraic logic De Morgan monoids Relevance logic Universal algebra Beth property UCTD |
| url | http://hdl.handle.net/2263/75178 |