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

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