Abstract
We show that an elementary class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular semigroups this allows an interpretation of a universal algebraic nature that is formulated entirely in terms of the associative binary operation of the semigroup, which serves as an alternative to the approach via so called e-varieties. In particular we prove that classes of Inverse semigroups, Orthodox semigroups, and E-solid semigroups are equational in our sense. We also examine which equations are valid in every semigroup.
Highlights
Groups may be characterized in terms of their binary operation alone as they form the class of semigroups that are both left and right simple, which is to say that a semigroup S is a group if and only if aS = Sa = S for all a ∈ S
A second observation is that G is a class of semigroups closed under the operations H and P, which are respectively the taking of homomorphic images, and the taking of direct products, but G is not closed under the taking of subsemigroups, so that G does not constitute a semigroup variety
It follows that the class of bisimple semigroups is closed under the taking of homomorphic images but, as we show, not under the taking of direct products
Summary
V of semigroups (a class closed under the operators H, P, and S, the taking of subalgebras) is, by Birkho 's theorem, dened by some countable set of identities, which are equations that may be expressed without the use of the existential symbol ∃. As K is dened by a family of equation systems it is an elementary class (denable in rst order logic) and is determined by its countably generated members As these coincide with HSP(K) it follows that HSP(K) = K. When the class of reducts of members of a variety is closed under H and E (as they are for groups and inverse semigroups), Theorem 3.1 shows that the class is denable by the equation systems.
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