A family of groups extending McLain's

Abstract

Given a strict partial order $\Delta$ on a set $\Lambda$ and an arbitrary ring $R$ with $1\ne 0$, the corresponding McLain group $M(\Delta )$ has been studied in depth. We construct a larger family of McLain groups $G(\Delta )$, where $\Delta$ is neither asymmetric nor transitive, while satisfying two weaker axioms. Structural properties common to all members~$G(\Delta )$ of this new family are investigated, including a group presentation, a description of the factors of its descending central series, a canonical form for its elements relative to any total order on~$\Delta$, and a recursive determination of its upper central series. In addition, we prove the natural isomorphism $G(\Delta )/G(\Gamma )\cong G(\Delta \setminus \Gamma )$, where $\Gamma$ is a normal subset $\Gamma$ of $\Delta$, and $G(\Gamma )$ and $G(\Delta \setminus \Gamma )$ are extended McLain groups on their own right. This result has no parallel in the classical context.