Equationally defined classes of semigroups

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathscr {C}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathscr {C}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is free of the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\forall $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∀</mml:mo> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\forall \cdots )(\exists \cdots )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>∀</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mo>∃</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> equation systems nor <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\exists \cdots )(\forall \cdots )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>∃</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>)</mml:mo> <mml:mo>(</mml:mo> <mml:mo>∀</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> systems suffice.</jats:p>