Definition:Set of Literals

From ProofWiki
Jump to navigation Jump to search

Informal Definition

Let $S$ be a set.

Informally, the set of literals on $S$ is obtained by adjoining formal inverses of the elements of $S$:

$S^\pm = S \cup \set {s^{-1} : s \in S}$

That is, for each $s \in S$ we add a formal symbol $s^{-1} \in S^\pm$ which we call the "formal inverse" but which is still just a symbol.

In particular $s^{-1}$ in the set of literals is not (yet) the actual inverse of $s$ under any algebraic operation.


Let $S$ be a set.

A set of literals on $S$ is a triple $\struct {S^\pm, \iota, \theta}$ 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 $\map \theta s = s^{-1}$

such that $S^\pm = \iota \sqbrk S \sqcup \theta \sqbrk {\iota \sqbrk S}$ is partitioned by the image of $S$ under $\iota$ and its image under $\theta$.

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

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

Let $\iota: S \to S^\pm$ be the canonical mapping:

$s \mapsto \tuple {s, 0}$

Let $\theta : S^\pm \to S^\pm$ be the mapping:

$\tuple {s, i} \mapsto \tuple {s, 1 - i}$

Also see