Infinite Convergent String-rewriting Systems and Cross-sections for Finitely Presented Monoids

F. OTTO; M. KATSURA; Y. KOBAYASHI

doi:10.1006/jsco.1998.0230

Infinite Convergent String-rewriting Systems and Cross-sections for Finitely Presented Monoids

Journal of Symbolic Computation ◽

10.1006/jsco.1998.0230 ◽

1998 ◽

Vol 26 (5) ◽

pp. 621-648 ◽

Cited By ~ 11

Author(s):

F. OTTO ◽

M. KATSURA ◽

Y. KOBAYASHI

Keyword(s):

Cross Sections ◽

Rewriting Systems ◽

String Rewriting ◽

Finitely Presented

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Polygraphs of finite derivation type

Mathematical Structures in Computer Science ◽

10.1017/s0960129516000220 ◽

2016 ◽

Vol 28 (2) ◽

pp. 155-201 ◽

Cited By ~ 6

Author(s):

YVES GUIRAUD ◽

PHILIPPE MALBOS

Keyword(s):

Word Problem ◽

Finiteness Condition ◽

Necessary Condition ◽

Finiteness Property ◽

Rewriting Systems ◽

Higher Dimensional ◽

String Rewriting ◽

Finite Derivation Type ◽

Finitely Presented ◽

Finite Presentations

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: Being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squier's results in the contemporary language of polygraphs and higher dimensional categories, with new proofs and relations between them.

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Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Fundamenta Informaticae ◽

10.3233/fi-2021-2002 ◽

2021 ◽

Vol 178 (3) ◽

pp. 173-185

Author(s):

Arthur Adinayev ◽

Itamar Stein

Keyword(s):

Isomorphism Problem ◽

Complete Characterization ◽

Hasse Diagram ◽

Subgraph Isomorphism ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting ◽

Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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