Left restriction monoids from left E-completions

Given a monoid S with E any non-empty subset of its idempotents, we present novel one-sided version idempotent completion call left E-completion. In general, the construction yields variant small category called constellation by Gould and Hollings. Under certain conditions, this is inductive, meaning that partial multiplication may be extended to give restriction semigroup, type unary semigroup whose operation models domain. We study properties those pairs S,E for which happens, characterise semigroups arise as such E-completions their submonoid elements having domain 1. As first applications, decompose functions on set X right total partitions respectively transformation TX X, binary relations under demonic composition E-completion left-total relations. many cases, including these three examples, embeds in Zappa-Szép product.