Axiomatizability in inclusive equational logics

摘要: A categorical framework for equational logics is presented, together with axiomatizability results in the style of Birkhoff. The distinctive structures used are inclusion systems, which an alternative to factorization systems required be unique rather than ‘up isomorphism’. In this framework, models any objects, and equations special epimorphisms C, while satisfaction injectivity. first result says that equations-as-epimorphisms define exactly quasi-varieties, suggesting actually represent conditional equations. fact, it shown projectivitysfreeness domain what makes difference between unconditional equations, defining varieties, as expected. An abstract version axiom choice seems sufficient free objects projective, case definitional power projective domain, respectively, same. Connections other formulations investigated, organization our logic institution.