Equationally defined classes of semigroups

Abstract We apply, in the context of semigroups, the main theorem from the authors’ paper “Algebras defined by equations” (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class $${\mathscr {C}}$$ C of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of $${\mathscr {C}}$$ C is free of the $$\forall $$ ∀ quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither $$(\forall \cdots )(\exists \cdots )$$ ( ∀ ⋯ ) ( ∃ ⋯ ) equation systems nor $$(\exists \cdots )(\forall \cdots )$$ ( ∃ ⋯ ) ( ∀ ⋯ ) systems suffice.