Neutral element for a multiary operation

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Definition

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.