# Looking for standard and consistent notation/terminology for the finite sequences/heaps on a set

Question 1. Does anyone know of standard and consistent notation for the following?

1. The set of non-empty finite sequences on a set.
2. As above, but including the empty sequence.
3. The set of finite non-empty heaps on a set.
4. As above, but including the empty heap.

Motivation. I'd like the following propositions to look similar and consistent.

Let $\mathrm{seq}(X)$ and $\mathrm{seq}_0(X)$ denote the finite sequences on $X$, where the former does not include the empty sequence but the latter does. Furthermore, let $\oplus$ denote concatenation.

Similarly, let $\mathrm{hea}(X)$ and $\mathrm{hea}_0(X)$ denote the finite heaps on $X$, where the former does not include the empty heap but the latter does. Furthermore, let $\uplus$ denote the heapic sum.

Then we have

1. An operation $*$ on $X$ is associative iff there exists a function $f : \mathrm{seq}(X) \rightarrow X$ satisfying the identity $f(a \oplus b) = f(a) * f(b).$

2. An operation $*$ on $X$ is associative and has an identity element iff there exists a function $f : \mathrm{seq}_0(X) \rightarrow X$ satisfying the identity $f(a \oplus b) = f(a) * f(b).$

3. An operation $*$ on $X$ is associative and commutative iff there exists a function $f : \mathrm{hea}(X) \rightarrow X$ satisfying the identity $f(A \uplus B) = f(A) * f(B).$

4. An operation $*$ on $X$ is associative and commutative and has an identity element iff there exists a function $f : \mathrm{hea}_0(X) \rightarrow X$ satisfying the identity $f(A \uplus B) = f(A) * f(B).$

Granted I have yet to sit down and actually prove these but they seem intuitively obvious.

Question 2. Is there a traditional name for the function $\mathrm{seq}_0 \rightarrow \mathrm{hea}_0$ that forgets order?

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