Affine parts of algebraic theories (Q788099)

For any finitary algebraic theory \(T\), let \(T^*\) be the coproduct of \(T\) and \(P\), where \(P\) is the theory of pointed sets, and let \(aT\) be the affine part of \(T\) in the sense of \textit{J. R. Isbell}, \textit{M. I. Klun}, and \textit{S. H. Schanuel} [Can. J. Math. 30, 231--237 (1978; Zbl 0362.18010)]. For any variety of algebras \(M\), let \(T_ M\) be the theory of \(M\). The author considers the following two conditions: (1) \(T\) is isomorphic with \((aT)^*\), (2) \(aT\) is isomorphic with \(a((aT)^*)\). It is shown that these conditions characterize varieties of modules within some classes of varieties. In particular, the following results are proved. Proposition 5. If a theory \(T\) satisfies (2) and has a subtheory \(V\) whose algebras are at least loops, then these algebras are in fact abelian groups. Proposition 7. Let \(M\) be a variety of groups. Then the following are equivalent: (i) \(T_ M\) satisfies (1), (ii) \(T_ M\) satisfies (2), (iii) \(M\) is a variety of abelian groups. Proposition 9. Let M be a variety of quasimodules over a ring \(R\). Then \(T_ M\) satisfies (2) if and only if \(M\) is a variety of \(R\)-modules.