This document defines a minimal algebraic structure derived from the symbolic transformation rule:
(∘(in)(᛫) = ¬(᛫))
This rule parses as follows: the operator ∘(in) acts as an inversion-context application. When applied to an undifferentiated placeholder operand (᛫), it yields its logical negation, ¬(᛫).
This encodes the meta-rule:
We now expand this into a formal algebra
- Let
$\mathcal{U}$ be a set of primitive tokens (e.g.,${᛫, a, b, \dots}$ ). - The rule is defined as:
$$∘_{in}(x) = \neg x, \quad \forall x \in \mathcal{U}$$ -
Closure: If
$x \in \mathcal{U}$ , then$\neg x \in \mathcal{U}$ .
- For composition:
$$∘_{in}(∘_{in}(x)) = ∘_{in}(\neg x) = \neg(\neg x) = x$$ - Thus,
$∘_{in}$ is self-inverse:$$∘_{in} \circ ∘_{in} = id$$ -
Involution holds directly:
$$\neg(\neg x) = x$$
-
$\mathcal{A}$ forms a Boolean-like involutive algebra with a single unary operator. - The operator set
${id, ∘_{in}}$ constitutes a cyclic group of order 2 under composition.
-
Identity:
$∘_{in}^0(x) = x$ -
Negation:
$∘_{in}^1(x) = \neg x$ -
Periodicity:
$∘_{in}^{2n}(x) = x$ , and$∘_{in}^{2n+1}(x) = \neg x$ for integer$n \geq 0$ .
-
Closure:
$\forall x \in \mathcal{U}, \neg x \in \mathcal{U}$ . -
Involution:
$\neg(\neg x) = x$ . -
Identity Pairing: Each element
$x$ is uniquely paired with$\neg x$ . -
No Fixed Points (non-degenerate case):
$x = \neg x$ implies$x$ is a self-dual element (introducing a degenerate case).
- Each orbit of
$∘_{in}$ has size 2:${x, \neg x}$ . - If a self-dual element exists, its orbit has size 1.
- The system is trivially associative due to its unary nature.
This algebra provides a foundational framework for exploring negation, symmetry, and periodicity in formal symbolic systems. It can be extended by introducing binary operations (e.g., meet/join) to form a full Boolean algebra, or integrated into larger linguistic and logical frameworks.