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

TL;DR

This paper solves the word problem for ${ ext{LG}}$ local groups by transforming ${ ext{kappa}}$-terms into canonical forms, enabling effective identification of equivalent terms over this pseudovariety.

Contribution

It introduces a method to compute canonical forms for ${ ext{kappa}}$-terms over ${f LG}$ and establishes a basis of identities for the generated ${ ext{kappa}}$-variety.

Findings

Canonical forms distinguish different interpretations over ${f LG}$.

A set of ${ ext{kappa}}$-identities forms a basis for the ${ ext{kappa}}$-variety.

The procedure provides an effective solution to the ${ ext{kappa}}$-word problem.

Abstract

In this paper we study the $\kappa$-word problem for the pseudovariety ${\bf LG}$ 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 ${\bf LG}$. 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 ${\bf LG}$.