{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T20:45:37Z","timestamp":1773261937971,"version":"3.50.1"},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2009,6,1]],"date-time":"2009-06-01T00:00:00Z","timestamp":1243814400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Aust. Math. Soc."],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We introduce partial actions of <jats:italic>weakly left <jats:italic>E<\/jats:italic>-ample semigroups<\/jats:italic>, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left <jats:italic>E<\/jats:italic>-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation <jats:italic>\u03b1<\/jats:italic>\u21a6<jats:italic>\u03b1<\/jats:italic><jats:sup>+<\/jats:sup>, where <jats:italic>\u03b1<\/jats:italic><jats:sup>+<\/jats:sup> is the identity map on the domain of <jats:italic>\u03b1<\/jats:italic>. We investigate the construction of \u2018actions\u2019 from such partial actions, making a connection with the <jats:italic>FA<\/jats:italic>-<jats:italic>morphisms<\/jats:italic> of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schr\u00f6der, and by the second author, are to be extended appropriately to the case of weakly left <jats:italic>E<\/jats:italic>-ample semigroups, then we must construct not global actions, but so-called <jats:italic>incomplete actions<\/jats:italic>. In particular, we show that a partial action of a weakly left <jats:italic>E<\/jats:italic>-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.<\/jats:p>","DOI":"10.1017\/s1446788708000542","type":"journal-article","created":{"date-parts":[[2009,8,4]],"date-time":"2009-08-04T14:00:04Z","timestamp":1249394404000},"page":"355-377","source":"Crossref","is-referenced-by-count":26,"title":["PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT <i>E<\/i>-AMPLE SEMIGROUPS"],"prefix":"10.1017","volume":"86","author":[{"given":"VICTORIA","family":"GOULD","sequence":"first","affiliation":[]},{"given":"CHRISTOPHER","family":"HOLLINGS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2009,6,1]]},"reference":[{"key":"S1446788708000542_ref28","doi-asserted-by":"publisher","DOI":"10.1007\/BF01361970"},{"key":"S1446788708000542_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(84)90092-6"},{"key":"S1446788708000542_ref21","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1977.68.161"},{"key":"S1446788708000542_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01470756"},{"key":"S1446788708000542_ref7","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300012973"},{"key":"S1446788708000542_ref5","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/28.3.285"},{"key":"S1446788708000542_ref29","doi-asserted-by":"publisher","DOI":"10.1007\/BF01351670"},{"key":"S1446788708000542_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s002330010032"},{"key":"S1446788708000542_ref23","doi-asserted-by":"publisher","DOI":"10.1142\/9789812702616_0012"},{"key":"S1446788708000542_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s002330010144"},{"key":"S1446788708000542_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(91)90242-Z"},{"key":"S1446788708000542_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-006-0665-7"},{"key":"S1446788708000542_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2004.11.006"},{"key":"S1446788708000542_ref4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-98-04575-4"},{"key":"S1446788708000542_ref16","unstructured":"[16] Jackson M. and Stokes T. , \u2018Algebras of partial maps\u2019, Proceedings of the Special Interest Meeting on Semigroups and Related Mathematics, University of Sydney, 2005, to appear."},{"key":"S1446788708000542_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-005-0558-1"},{"key":"S1446788708000542_ref30","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(89)90054-6"},{"key":"S1446788708000542_ref24","first-page":"91","article-title":"Restrictively multiplicative algebras of transformations","volume":"4","author":"Schein","year":"1970","journal-title":"Izv. Vys\u0161. U\u010debn. Zaved. Mat."},{"key":"S1446788708000542_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02572784"},{"key":"S1446788708000542_ref31","first-page":"71","article-title":"Menger\u2019s function systems","volume":"11","author":"Trokhimenko","year":"1973","journal-title":"Izv. Vys\u0161. U\u010debn. Zaved. 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