{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T14:14:55Z","timestamp":1649168095326},"reference-count":33,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2003,12]]},"abstract":"<jats:p>A theory of the semidirect product of categories and the derived category of a category morphism is presented. In order to include division (\u227a) in this theory, the traditional setting of these constructions is expanded to include relational arrows. In this expanded setting, a relational morphism \u03c6 : M \u2192 N of categories determines an optimal decomposition [Formula: see text] where [Formula: see text] denotes semidirect product and D(\u03c6) is the derived category of \u03c6.<\/jats:p><jats:p>The theory of the semidirect product of varieties of categories, V * W, is developed. Associated with each variety V of categories is the collection [Formula: see text] of relational morphisms whose derived category belongs to V. The semidirect product of varieties and the composition of classes of the form [Formula: see text] are shown to stand in the relationship [Formula: see text] The associativity of the semidirect product of varieties follows from this result.<\/jats:p><jats:p>Finally, it is demonstrated that all the results in the article concerning varieties of categories have pseudovariety and monoidal versions. This allows us to furnish a straightforward proof that [Formula: see text] for both varieties and pseudovarieties of monoids.<\/jats:p>","DOI":"10.1142\/s021819670300150x","type":"journal-article","created":{"date-parts":[[2004,1,27]],"date-time":"2004-01-27T09:43:48Z","timestamp":1075196628000},"page":"627-703","source":"Crossref","is-referenced-by-count":12,"title":["CATEGORIES AS ALGEBRA, II"],"prefix":"10.1142","volume":"13","author":[{"given":"BENJAMIN","family":"STEINBERG","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada"}]},{"given":"BRET","family":"TILSON","sequence":"additional","affiliation":[{"name":"14079 Lower Colfax Rd, Grass Valley, CA 95945, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(89)90124-2"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1142\/9789812831644"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196791000079"},{"key":"rf4","volume-title":"Category Theory for Computer Science","author":"Barr M.","year":"1995"},{"key":"rf5","first-page":"413","volume":"25","author":"Brown R.","journal-title":"Proc. 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