{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T06:44:35Z","timestamp":1740120275877,"version":"3.37.3"},"reference-count":11,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"crossref","award":["306534\/2016-9"],"award-info":[{"award-number":["306534\/2016-9"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004901","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de Minas Gerais","doi-asserted-by":"crossref","award":["APQ-01149-18"],"award-info":[{"award-number":["APQ-01149-18"]}],"id":[{"id":"10.13039\/501100004901","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:p> Let [Formula: see text] be an algebraically closed field of characteristic zero and let [Formula: see text] be a minimal superalgebra with semisimple part [Formula: see text], where each [Formula: see text] is a simple superalgebra. We say that [Formula: see text] is [Formula: see text]-special if [Formula: see text] and there exists [Formula: see text] such that [Formula: see text], with [Formula: see text], whereas, for the remaining indices [Formula: see text], [Formula: see text]. In [1], Avelar et al. classified the [Formula: see text]-special minimal superalgebras and described some necessary and sufficient conditions for the varieties generated by such superalgebras to be minimal of [Formula: see text]-exponent [Formula: see text]. In this paper, we give continuity to a such study and we made explicit that the [Formula: see text]-gradings play an important role in the minimality of some varieties. We provide sufficient conditions, depending on the [Formula: see text]-grading of [Formula: see text], for the variety generated by [Formula: see text] to be minimal. Moreover, we characterize two families of [Formula: see text]-special minimal superalgebras and we give necessary and sufficient conditions (depending on their [Formula: see text]-gradings) for the varieties generated by them to be minimal. <\/jats:p>","DOI":"10.1142\/s0218196723500145","type":"journal-article","created":{"date-parts":[[2022,12,9]],"date-time":"2022-12-09T03:19:45Z","timestamp":1670555985000},"page":"261-288","source":"Crossref","is-referenced-by-count":0,"title":["\u21242-gradings and minimal varieties of \u21242-graded PI algebras"],"prefix":"10.1142","volume":"33","author":[{"given":"Danilo Vilela","family":"Avelar","sequence":"first","affiliation":[{"name":"Departamento de An\u00e1lise, Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade Federal Fluminense, 24210-201 NITER\u00d3I, Brazil"}]},{"given":"Viviane Ribeiro Tomaz","family":"da Silva","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto de Ci\u00eancias Exatas, Universidade Federal de Minas Gerais, 30161-970 BELO HORIZONTE, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2023,1,19]]},"reference":[{"key":"S0218196723500145BIB001","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1080\/00927872.2019.1659287","volume":"48","author":"Avelar D. 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