Definition:Set of Literals

Informal definition
Let $S$ be a set.

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

Definition
Let $S$ be a set.

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

such that $S^\pm = \iota(S) \sqcup \theta(\iota(S))$ is the disjoint union of the image of $S$ under $\iota$ and its image under $\theta$.

Explicitly, $S^\pm$ can be defined as follows.

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

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

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

Also see

 * Definition:Group Word on Set