The word problem for $\kappa$-terms over the pseudovariety of local groups

In this paper we study the $\kappa$-word problem for the pseudovariety ${\bf LG}$oflocalgroups,where$\kappa$ is the canonical signature consisting of the multiplication and the pseudoinversion. We solve this problem by transforming each arbitrary $\kappa$-term $\alpha$ into another one called the canonical form of $\alpha$ and by showing that different canonical forms have different interpretations over $\mathbf{L}\mathbf{G}$. The procedure of construction of these canonical forms consists in applying elementary changes determined by a certain set $\mathrm{\Sigma }$ of $\kappa$-identities. As a consequence, $\mathrm{\Sigma }$ is a basis of $\kappa$-identities for the $\kappa$-variety generated by $\mathbf{L}\mathbf{G}$.