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Locality semigroups were proposed recently as one of the basic localityalgebraic structures, which are studied in mathematics and physics. Pathsemigroups and partial semigroups were also developed by many authors in theliterature. In this paper, we study free objects in the category of refinedlocality semigroups. It turns out that the path locality semigroup of a quiveris the free refined locality semigroup on a locality set. We also explore therelationships among locality semigroups, partial semigroups and path localitysemigroups, concluding that the path locality semigroup is a proper subclass ofthe intersection of locality semigroups and partial semigroups. In particular,the class of refined locality semigroups is a proper subclass of stronglocality semigroups. Furthermore, we show that, when a partial semigroup is arefined locality semigroup, one can extend it a strong semigroup with zero.

Several locality semigroups, path semigroups and partial semigroups