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

Abstract

In this paper we study the $\kappa$-word problem for the pseudovariety $\mathbf{LG}$ of local groups, 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 $\mathbf{LG}$. The procedure of construction of these canonical forms consists in applying elementary changes determined by a certain set $\Sigma$ of $\kappa$-identities. As a consequence, $\Sigma$ is a basis of $\kappa$-identities for the $\kappa$-variety generated by $\mathbf{LG}$.