 Finally, we can say that m is compositional simpliciter just in case m is F-compositional for each syntactic operation in E. Whenever m is compositional, it induces the semantic algebra \(\mathbf{M} = \langle M, (G_{\gamma})_{\gamma \in \Gamma}\rangle\) on M, and it is a homomorphism between E and M; cf. Westerståhl (1998). (For details, variants, and formal results, see Janssen (1983, 1997), Hodges (2001), and Pagin &amp; Westerståhl (2010a). For generalizations that cover languages with various sorts of context-dependence, see Pagin (2005), Pagin &amp; Pelletier (2007), and Westerståhl (2012).)