Is there a narrower monoidal structure definition?

Question

A monoid has a binary operation which is $A\times A\to A$.

This lets us to repeat the same operation for arbitrarily large finite number of elements, like taking sum of a long list of numbers, since our domain and codomain values are of the same type.

So if we think of a list of numbers like 1, 2, 3, 4 and use + operation as binary operator to accumulate the sum:

(left function parameter is used as accumulator, starts from 0 which is identity element for +)

(1) $0 + 1 = 1$

(2) $1 + 2 = 3$

(3) $3 + 3 = 6$

(4) $6 + 4 = 10$

What i realized is; the domain and codomain of the binary operation actually doesn't need to be of the same type to supply this behavior.

As an example if we have a piecewise function, and that function returns

• two times the number if it is odd (as an example 3 becomes 6)

• keep the value if it is even (as an example 4 returns 4)

And we use this piecewise function to form our binary operation

$Even\times Number\to Even$

We are still able to reuse the function over and over again to take sum of a long list.

(1) $0 + 1*2 = 2$

(2) $2 + 2 = 4$

(3) $4 + 3*2 = 10$

(4) $10 + 4 = 14$

So my question is, is there a name for this narrower definition of a monoid-like structure that uses this kind of a binary operation definition ($A\times B\to A$, $A\subseteq B$) instead of ($A\times A\to A$)?

Answer 1

A binary operation $A\times B\to A$ (without your requirement that $A\subseteq B$) is sometimes called an action (of the set $B$ on the set $A$ on the right). Similarly, a binary operation $B\times A\to A$ would be an action of $B$ on $A$ on the left.

If $(B,\cdot,e)$ is itself a monoid, then we might additionally want to require that an action $*\colon B\times A\to A$ respect the monoid operation, in the sense that

• $e*a = a$ for all $a\in A$.
• $b*(b'*a) = (b\cdot b')*a$ for all $b,b'\in B$ and $a\in A$.

If these axioms are satisfied, we call the action a monoid action (of the monoid $B$ on the set $A$ on the left). There are obvious dual axioms for monoid actions on the right.

The monoid action axioms are the analogues of the identity and associativity axioms for monoids, and they ensure that given a list $b_1b_2\dots b_k a$ with $k\in \mathbb{N}$, $b_i\in B$ for all $i$, and $a\in A$, there is a unique way to "multiply up" this list.