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 ${\mathrm{\Sigma }}_m^K$ 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 ${\mathrm{\Sigma }}_m^K$. In general, $C_m^K$ is a proper subclass of $V({\mathrm{\Sigma }}_m^K)$, the variety of algebras defined by ${\mathrm{\Sigma }}_m^K$.

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 $V({\mathrm{\Sigma }}_m^K)$. We show that the free algebras of $V({\mathrm{\Sigma }}_m^K)$ correspond to $K$-perfect $m$-cycle systems, so $C_m^K$ generates $V({\mathrm{\Sigma }}_m^K)$. We also answer two questions asked in [5] concerning subvarieties of $V({\mathrm{\Sigma }}_m^K)$. Many of these results can be unified in the result that for any subset $K^{\prime }$ of $K$, $V({\mathrm{\Sigma }}_m^{K^{\prime }})$ is generated by the class of algebras corresponding to finite $K$-perfect $m$-cycle systems.