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Boone group presentation and special word

Definition

Use the finite positive presentation $Γ$ and its relation notation $F_{i} q_{a (i)} G_{i} = H_{i} q_{b (i)} K_{i}$ from Boone machine semigroup and augmented configurations. All generator alphabets below are disjointly tagged.

For a signed tape word $U = s_{1}^{e_{1}} \dots s_{m}^{e_{m}}$ , $e_{j} \in {1, - 1}$ , set $U^{#} = s_{1}^{- e_{1}} \dots s_{m}^{- e_{m}}$ . Thus $(U V)^{#} = U^{#} V^{#}$ , $(U^{#})^{#} = U$ , and $ε^{#} = ε$ . A cancellation pair is carried to a cancellation pair; applying $#$ twice recovers the original pair. Consequently this operation is well-defined on free tape words modulo free cancellation. It is neither reversal nor a claimed operation on the semigroup quotient.

Define $B$ by generators $\overset{ˉ}{S} ⊔ \overset{ˉ}{Q} ⊔ {r_{i} : i \in I} ⊔ {x, t, k}$ and the following relations for every $s \in \overset{ˉ}{S}$ and $i \in I$ : $x s = s x^{2}, r_{i} s = s x r_{i} x,$ $r_{i}^{- 1} (F_{i}^{#} q_{a (i)} G_{i}) r_{i} = H_{i}^{#} q_{b (i)} K_{i},$ $t r_{i} = r_{i} t, t x = x t, k r_{i} = r_{i} k, k x = x k,$ $k (q^{- 1} t q) = (q^{- 1} t q) k .$ Each displayed equation $u = v$ means the relator $u v^{- 1}$ in the free-group quotient of Recursive presentations and finite presentations of groups. Both the generating set and relator list are finite. There are no other defining equations, in particular none commuting state letters with $x$ .

A special word is the spelled word $Σ = X^{#} q_{j} Y$ , where $X, Y \in {\overset{ˉ}{S}}^{*}$ are positive tape words (empty is allowed) and $q_{j} \in \overset{ˉ}{Q}$ . Set $Σ^{*} = X q_{j} Y$ , a nonempty positive semigroup word, and $W (Σ) = k Σ^{- 1} t Σ k^{- 1} Σ^{- 1} t^{- 1} Σ .$ The operation $#$ is on the tape words in this spelling, not on arbitrary elements of $B$ . For empty tape contexts, $Σ = q_{j}$ and $Σ^{*} = q_{j}$ .

Source locator

Rotman, Chapter 12, printed pp.430–431, sharp notation and the displayed Boone presentation. Inverses in $W$ reverse the full word; sharp preserves the order of its tape letters.

Depends on

Used by

Boone sharp is not word inversion Example
Boone base groups and associated free bases Lemma
Boone hnn tower and auxiliary subgroups Lemma
Boone positive history pushing Lemma
Boone positive history reconstruction Lemma
Novikov boone undecidability of the word problem Theorem

Dependency tree · two levels

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Sources

Joseph J. Rotman, An Introduction to the Theory of Groups, Chapter 12, pp.430–431, sharp notation and Boone presentation (standard reference, not scraped)