Neutral element for a multiary operation

All-sided neutral element
Suppose $$S$$ is a set and $$f:S^n \to S$$ is a $$n$$-ary operation for $$f$$. An element $$e \in S$$ is termed a neutral element for $$f$$ if the following holds: $$f$$ evaluated at any $$n$$-tuple where $$n - 1$$ of the entries are equal to $$e$$ and the remaining entry is $$a \in S$$, gives output $$a$$. This is true regardless of where we place $$a$$ and is also true if $$a = e$$.

The term neutral element, when used without qualification, is used in the context $$n = 2$$, i.e., for a binary operation, i.e., a magma.

Left neutral element
Suppose $$S$$ is a set and $$f:S^n \to S$$ is a $$n$$-ary operation for $$f$$. An element $$e \in S$$ is termed a left neutral element for $$f$$ if the following holds: $$f(e,e,e,\dots,e,a) = a$$ for all $$a \in S$$.

Right neutral element
Suppose $$S$$ is a set and $$f:S^n \to S$$ is a $$n$$-ary operation for $$f$$. An element $$e \in S$$ is termed a right neutral element for $$f$$ if the following holds: $$f(a,e,e,e,\dots,e) = a$$ for all $$a \in S$$.

Neutral element for a given position
Suppose $$S$$ is a set and $$f:S^n \to S$$ is a $$n$$-ary operation for $$f$$. Suppose $$i \in \{ 1,2,\dots,n \}$$. An element $$e \in S$$ is termed a neutral element for position $$i$$ if $$f$$ if the following holds: $$f(e,\dots,e,a,e,e,\dots,e) = a$$ for all $$a \in S$$ where $$a$$ appears in the $$i^{th}$$ position.

Facts
The case $$n = 2$$ is special because we can deduce equality of left and right neutral element and therefore also deduce that binary operation on magma determines neutral element. For higher $$n$$, there could be more than one neutral element.