On countably generated locally $\mathfrak M$-algebras

We show how to construct countably generated locally nilpotent groups, rings, and algebras, locally finite groups, rings, and algebras over a finite field, and other countably generated universal algebras possessing certain properties locally. The construction possesses a property close to universality. For example, with each function $f\colon N\to N$ defined on the natural numbers $N$ and assuming values in $N$ there is associated a countably generated locally nilpotent algebra $\mathscr L(f)$. If $f$ is an unbounded increasing function, then any countably generated or finitely generated locally nilpotent algebra $R$ is a homomorphic image of $\mathscr L(f)$. On the other hand, if $f$ and $g$ are any two increasing functions, then $\mathscr L(f)$ and $\mathscr L(g)$ are isomorphic if and only if $f$ and $g$ agree.
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