{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T12:43:21Z","timestamp":1749645801706},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,2,25]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we investigate the variety <jats:bold>RDP<\/jats:bold> of regular double <jats:italic>p<\/jats:italic>-algebras and its subvarieties <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>\n                     <jats:italic>n<\/jats:italic>\n                  <\/jats:sub>, <jats:italic>n<\/jats:italic> \u2265 1, of range <jats:italic>n<\/jats:italic>. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub> and show that this variety is locally finite. We also show that the lattice of subvarieties of <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub>, <jats:italic>L<jats:sub>V<\/jats:sub>\n                  <\/jats:italic>(<jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub>), is isomorphic to the lattice of down sets of the poset {1} \u2295 (\u2115 \u00d7 \u2115). We describe all the subvarieties of <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub> and conclude that <jats:italic>L<jats:sub>V<\/jats:sub>\n                  <\/jats:italic>(<jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub>) is countably infinite. An equational basis for each proper subvariety of <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>1<\/jats:sub> is given. To study the subvarieties <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>\n                     <jats:italic>n<\/jats:italic>\n                  <\/jats:sub> with <jats:italic>n<\/jats:italic> \u2265 2, Priestley duality as it applies to regular double <jats:italic>p<\/jats:italic>-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2<jats:sup>\n                     <jats:italic>\u2135<\/jats:italic>\n                     <jats:sub>0<\/jats:sub>\n                  <\/jats:sup>. We also use Priestley duality to prove that <jats:bold>RDP<\/jats:bold> and each of its subvarieties <jats:bold>RDP<\/jats:bold>\n                  <jats:sub>\n                     <jats:italic>n<\/jats:italic>\n                  <\/jats:sub> are generated by their finite members.<\/jats:p>","DOI":"10.1515\/ms-2017-0200","type":"journal-article","created":{"date-parts":[[2019,1,23]],"date-time":"2019-01-23T18:31:03Z","timestamp":1548268263000},"page":"15-34","source":"Crossref","is-referenced-by-count":7,"title":["Regular double <i>p<\/i>-algebras"],"prefix":"10.1515","volume":"69","author":[{"given":"M. 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