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It is structurally complete if and only if the free \u2135<jats:sub>0<\/jats:sub>\u2010generated algebra in  can serve as <jats:bold><jats:italic>A<\/jats:italic><\/jats:bold>. A consequence of this demand, called \u2018passive structural completeness\u2019 (PSC), is that the nontrivial members of  all satisfy the same existential positive sentences. We prove that if  is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if  is relatively semisimple then it is generated by one \u2010simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance\u00a0logics.<\/jats:p>","DOI":"10.1002\/malq.201900012","type":"journal-article","created":{"date-parts":[[2020,6,22]],"date-time":"2020-06-22T06:28:30Z","timestamp":1592807310000},"page":"150-172","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Singly generated quasivarieties and residuated structures"],"prefix":"10.1002","volume":"66","author":[{"given":"Tommaso","family":"Moraschini","sequence":"first","affiliation":[{"name":"Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vod\u00e1renskou v\u011b\u017e\u00ed 2, 182 07 Prague 8 Czech Republic"}]},{"given":"James G.","family":"Raftery","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics University of Pretoria Private Bag X20, Hatfield Pretoria 0028 South Africa"},{"name":"DST\u2010NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE\u2010MaSS) 1 Jan Smuts Avenue, Braamfontein 2000 Johannesburg South Africa"}]},{"given":"Johann J.","family":"Wannenburg","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics University of Pretoria Private Bag X20, Hatfield Pretoria 0028 South Africa"},{"name":"DST\u2010NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE\u2010MaSS) 1 Jan Smuts Avenue, Braamfontein 2000 Johannesburg South Africa"}]}],"member":"311","published-online":{"date-parts":[[2020,6,21]]},"reference":[{"key":"e_1_2_12_2_1","volume-title":"Entailment: The Logic of Relevance and Necessity","author":"Anderson A. 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