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Denote by PSelf Ω (resp., Self Ω) the partial (resp., full) transformation monoid over a set Ω, and by Sub V (resp., End V ) the collection of all subspaces (resp., endomorphisms) of a vector space V . We prove various results that imply the following: (1) If card Ω 2, then Self Ω has a semigroup embedding into the dual of Self Γ iff card Γ 2 card Ω . In particular, if Ω has at least two elements, then there exists no semigroup embedding from Self Ω into the dual of PSelf Ω. (2) If V isdoi:10.4064/fm202-2-2 fatcat:jm7yw446ibcshcel6nvfylm4xe