Definition:Set of Literals

Informal definition
Let $X$ be a set.

Informally, the set of literals on $X$ is obtained by adjoining formal inverses of the elements of $X$:
 * $X^\pm = X \cup \{x^{-1} : x \in X\}$

Definition
Let $X$ be a set.

A set of literals on $X$ is a triple $(X^\pm, \iota, \theta)$ where:
 * $X^\pm$ is a set
 * $\iota : X \to X^\pm$ is a mapping, the canonical injection
 * $\theta : X^\pm \to X^\pm$ is an involution without fixed points, the inversion mapping, and we also denote $\theta(x) = x^{-1}$

such that $X^\pm = \iota(X) \sqcup \theta(\iota(X))$ is the disjoint union of the image of $X$ under $\iota$ and its image under $\theta$, that can be defined as follows:

Let $X^\pm = X \sqcup X = X \times \{0\} \cup X \times \{1\}$ be the disjoint union of $X$ with $X$.

Let $\iota : X \to X^\pm$ be the canonical mapping:
 * $x \mapsto (x, 0)$

Let $\theta : X^\pm \to X^\pm$ be the mapping:
 * $(x, i) \mapsto (x, 1-i)$

Also see

 * Definition:Group Word on Set