Abstract
We first consider the ternary semigroup of mappings between two nonempty sets $$X$$ and $$Y$$ , that is $$T[X,Y]$$ . Then we characterize different types of elements in $$T[X,Y]$$ . In particular, we provide some necessary and sufficient conditions for an element of $$T[X,Y]$$ to become a left regular, a right regular, a completely regular and an idempotent element of the ternary semigroup $$T[X,Y]$$ . Our results enrich the structure of the ternary semigroup of mappings.
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