# Name for generalization of property: $f^n(x) \ne x$ for all $n > 0$

I am curious about how to specify with standard terminology that a certain function is non-periodic, in the following sense:

In the simple case of a unary operation $$f: X \to X$$, this property would specify that, for all $$x \in X$$:

1. $$f(x) \ne x$$
2. $$f(f(x)) \ne x$$
3. $$f(f(f(x))) \ne x$$
...

For this case, I believe I could simply say that "$$f$$ has no periodic orbit".

However, the case I am actually curious about is that of a binary operation $$f: X \times X \to X$$. In this case, the property would specify that, for all $$a, b \in X$$:

1. $$f(a, b) \notin \{a, b\}$$
2. $$\{f(f(a, b), c), f(c, f(a, b))\} \cap \{a, b\} = \varnothing$$
3. $$\{f(f(f(a, b), c), d), f(f(c, f(a, b)), d), f(d, f(f(a, b), c)), f(d, f(c, f(a, b)))\} \cap \{a, b\} = \varnothing$$
...

An example of a binary operation that exhibits this property would be the addition of positive integers, i.e., $$+: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$$, since, for all $$a, b \in \mathbb{Z}^+$$:

1. $$(a + b) \notin \{a, b\}$$
2. $$(a + b + c) \notin \{a, b\}$$
3. $$(a + b + c + d) \notin \{a, b\}$$
...

More generally, I could formally specify this property for any $$n$$-ary operation $$f : X^n \to X$$ as follows: $$\forall x \in X, k \in \mathbb{Z}^+ : x \notin F_k(x),$$

where $$F_0(x) = \{x\}$$ $$F_{k+1}(x)=\{f(\mathbf v) : \mathbf v \in X^n \land \exists i:\mathbf v_i \in F_k(x)\}$$

So my question is: does this property have a name? I'd prefer to just call it by its name, if there exists a standard term for it, rather than formally specifying it (e.g., let $$f$$ be a [fill-in-the-blank] binary operator on $$X$$).

• Another way to say it (I think): if $\rightarrow$ is the smallest relation satisfying $x \rightarrow f(a_1, \dotsc, a_{i - 1}, x, a_{i + 1}, \dotsc, a_n)$ for all $i$ and all $x, a_1, \dotsc, a_n \in X$, then the transitive closure $\Rightarrow$ of $\rightarrow$ is anti-reflexive. – LSpice Nov 20 '18 at 18:42
• I think it captures your idea to think of the binary (or $n$-ary) operation as a multi-valued single-place function, in the sense I've discussed above, and then declare that the multi-valued function is strongly non-periodic, in the sense that none of the possible orbits through an element revisits its starting point. – LSpice Nov 20 '18 at 18:44