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 $X$).

  • 3
    $\begingroup$ 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. $\endgroup$ – LSpice Nov 20 '18 at 18:42
  • $\begingroup$ 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. $\endgroup$ – LSpice Nov 20 '18 at 18:44

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