{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T22:01:47Z","timestamp":1699999307025},"reference-count":20,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:p>We study the class of symmetric [Formula: see text]-ary bands. These are [Formula: see text]-ary semigroups [Formula: see text] such that [Formula: see text] is invariant under the action of permutations and idempotent, i.e., satisfies [Formula: see text] for all [Formula: see text]. We first provide a structure theorem for these symmetric [Formula: see text]-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong [Formula: see text]-ary semilattice of [Formula: see text]-ary semigroups and we show that the symmetric [Formula: see text]-ary bands are exactly the strong [Formula: see text]-ary semilattices of [Formula: see text]-ary extensions of Abelian groups whose exponents divide [Formula: see text]. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric [Formula: see text]-ary band to be reducible to a semigroup.<\/jats:p>","DOI":"10.1142\/s0218196721500478","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T07:47:22Z","timestamp":1630396042000},"page":"1319-1338","source":"Crossref","is-referenced-by-count":1,"title":["On the structure of symmetric n-ary bands"],"prefix":"10.1142","volume":"31","author":[{"given":"Jimmy","family":"Devillet","sequence":"first","affiliation":[{"name":"University of Luxembourg, Department of Mathematics, Maison du Nombre, 6, Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg"}]},{"given":"Pierre","family":"Mathonet","sequence":"additional","affiliation":[{"name":"University of Li\u00e8ge, Mathematics Research Unit, All\u00e9e de la D\u00e9couverte 12 - B37, B-4000 Li\u00e8ge, Belgium"}]}],"member":"219","published-online":{"date-parts":[[2021,8,30]]},"reference":[{"key":"S0218196721500478BIB002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1954-0062119-9"},{"key":"S0218196721500478BIB003","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/007.1"},{"key":"S0218196721500478BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-019-0626-0"},{"key":"S0218196721500478BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s13366-020-00551-2"},{"key":"S0218196721500478BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-018-9980-z"},{"key":"S0218196721500478BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2007.09.005"},{"key":"S0218196721500478BIB008","first-page":"163","volume":"14","author":"Dudek W. 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