The variety generated by the class of \(K\)-perfect \(m\)-cycle systems

Abstract

A method called the standard construction generates an algebra from a \(K\)-perfect \(m\)-cycle system. Let \({C}_m^K\) denote the class of algebras generated by \(K\)-perfect \(m\)-cycle systems. For each \(m\) and \(K\), there is a known set \(\Sigma_m^K\) of identities which all the algebras in \({C}_m^K\) satisfy. The question of when \({C}_m^K\) is a variety is answered in [2]. When \({C}_m^K\) is a variety, it is defined by \(\Sigma_m^K\). In general, \({C}_m^K\) is a proper subclass of \({V}(\Sigma_m^K)\), the variety of algebras defined by \(\Sigma_m^K\).

If the standard construction is applied to partial \(K\)-perfect \(m\)-cycle systems, then partial algebras result. Using these partial algebras, we are able to investigate properties of \({V}(\Sigma_m^K)\). We show that the free algebras of \({V}(\Sigma_m^K)\) correspond to \(K\)-perfect \(m\)-cycle systems, so \({C}_m^K\) generates \({V}(\Sigma_m^K)\). We also answer two questions asked in [5] concerning subvarieties of \({V}(\Sigma_m^K)\). Many of these results can be unified in the result that for any subset \(K’\) of \(K\), \({V}(\Sigma_m^{K’})\) is generated by the class of algebras corresponding to finite \(K\)-perfect \(m\)-cycle systems.