Author:
Higgins Peter M.,Jackson Marcel
Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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