Locally finite varieties with large free spectra. (Q1771920)

For a variety \(\mathcal V\), \(f_{\mathcal V}(n)\) denotes the size of the free algebra on \(n\) generators in \(\mathcal V\) and \(g_{\mathcal V}(n)\) is the number of non-isomorphic algebras in \(\mathcal V\) generated by \(n\) or fewer elements. Thus \(\mathcal V\) is locally finite if \(f_{\mathcal V}(n)\) is an integer for any positive integer \(n\). The aim of the paper is to show that there are no bounds for locally finite varieties: Let \(f\) be any function from \(N\) to \(N\). There exists a locally finite variety \(\mathcal V\) of groupoids such that \(f_{\mathcal V}(n) \geq f(n)\) and \(g_{\mathcal V}(n) \geq g(n)\) for all \(n > 0\). Let \(f\) be any recursive function. There exists a variety \(\mathcal V\) of groupoids which is locally finite and finitely axiomatizable such that \(f_{\mathcal V}(n) \geq f(n)\) and \(g_{\mathcal V}(n) \geq g(n)\) for all \(n > 0\). Let \(f\) be any function from \(N\) to \(N\). There exists a locally finite discriminator variety \(\mathcal V\) such that \(f_{\mathcal V}(n) > h(g_{\mathcal V}(n))\) for all \(n > 1\).