Does there exist a set $S$ and two binary operations $+$ and $*$ on that set, such that both the structures $(S;+)$ and $(S;*)$ have a finite basis of identities, but the conjoined structure $(S;+,*)$ does not have a finite basis? Also, what about vice versa? That is, both $(S;+)$ and $(S;*)$ are non-finitely based, but the structure $(S;+,*)$ is finitely based.
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I can answer the first question.
Every finite group $\langle G; \cdot\rangle$ is finitely based (see "Identical relations in finite groups". S. Oates and B. Powell. Journal of Algebra, 1964.) We don't need the unary operation $^{-1}$ or the constant $1$ in the signature since they are encoded by $x\mapsto x^{n-1}$ and $x\mapsto x^n$ where $n=\vert G\vert$.
It seems that every set $G$ with a constant binary operation $c(x,y)=c\in G$ is described by the identity $$c(x,y)\approx c(z,w).$$
However, Roger Bryant found a finite pointed group $\langle G; \cdot, c\rangle$ that is not finitely based (see "Laws of finite pointed groups". R. Bryant. Bulletin of the LMS, 1982.) It doesn't matter if we consider $c$ as a constant or a constant binary operation.
Eran
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3For the second question: Let $({0,1,2},+)$ be Murskii's groupoid and let $({0,1,2},)$ be Murskii's groupoid with the roles of $0$ and $2$ switched. The algebra $({0,1,2},+,)$ has a chain of Jonsson terms of length 5. – Keith Kearnes Jun 20 '21 at 07:33
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@KeithKearnes I had tried variations of that idea but gave up before I got the right combination. Thanks! – Eran Jun 20 '21 at 17:43