Generalized right central loops

Abstract

Four generalized identities corresponding to the four right central identities are introduced for the first time and studied in loops. It is shown that the four new identities are equivalent in a loop. The algebraic properties of a generalized right central loop (GRCL) \((G,\cdot ,\sigma )\) is investigated and some equivalent characterizing forms of the self map \(\sigma \) are found. The self map \(\sigma \) in a GRCL \((G,\cdot ,\sigma )\) is shown to be of order \(2k\) if and only if the kth power of some nuclear element is central. The self map \(\sigma \) in a GRCL \((G,\cdot ,\sigma )\) is shown to be of order \(2k+1\) if and only if the inverse mapping is the 2k+1th power of the right translation of some nuclear element. If \((G,\cdot ,\sigma )\) is a GRCL, then some equivalent necessary and sufficient conditions for \((G,\cdot ,\sigma ^k),~k\in \mathbb {Z}\) to be a GRCL are established. It is established that a loop \((G,\cdot ,\sigma )\) is a generalized central loop (GCL) if and only if it is a GRCL and a generalized left central loop (GLCL). It is also established that a loop \((G,\cdot ,\sigma )\) is a generalized central loop (GCL) if and only if it is a GRCL and a left inverse property loop (LIPL). It is likewise established that a loop \((G,\cdot ,\sigma )\) is a generalized central loop (GCL) if and only if it is a GLCL and a right inverse property loop (RIPL). An identity that is a necessary and sufficient condition for a GRL to be a group is established. A loop is shown to be a group if and only if it is a GRL and it obeys this identity. Finally, we attempt to construct a GRCL by using a group and an arbitrary subgroup of it. A family and a group of GRCLs are also constructed.