Translatability and translatable semigroups

The concept of a k-translatable groupoid is explored in depth.Some properties of idempotent k-translatable groupoids, ORG SPELT FLOUR left cancellative k-translatable groupoids and left unitary k-translatablegroupoids are proved.Necessary and sufficient conditions are found for a left cancellative k-translatable groupoid to be a semigroup.Any such

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