Abstract
Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product \({T\ltimes {\mathcal {Y}}}\) of a semilattice \({\mathcal {Y}}\) by a monoid T, where the action of T on \({\mathcal {Y}}\) is injective with images of the action being order ideals of \({\mathcal {Y}}\). Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form \({{\mathcal {Y}}\rtimes T}\).
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Research supported by the Hungarian National Foundation for Scientific Research grant no. K77409 and K83219, by the Hungarian National Development Agency grant no. TAMOP-4.2.1/B-09/1/KONV-2010-0005, and by EPSRC grant no. EP/I032312/1. The first author would like to thank Profs. P. N. Ánh, László Márki and Mária Szendrei for their hospitality during her visits to Hungary in 2011.
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Gould, V., Szendrei, M.B. Proper restriction semigroups – semidirect products and W-products. Acta Math Hung 141, 36–57 (2013). https://doi.org/10.1007/s10474-013-0322-z
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DOI: https://doi.org/10.1007/s10474-013-0322-z