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A Noetherian and confluent rewrite system for idempotent semigroups


Let B be a semigroup with the additional relation

$$\begin{gathered} xx \Rightarrow x \hfill \\ xyz \Rightarrow xz if x \mathop {CI}\limits_ = z and xy\mathop {CI}\limits_ = z \hfill \\ \end{gathered} $$

B is called aband or anidempotent semigroup [3].

It is shown in this paper that the replacement rules (rewrites) resulting from the axiom of idempotence:

$$\forall w \in B.ww = w$$

can be replaced by theNoetherian, confluent, conditional rewrites (i. e. a terminating replacement system having the Church-Rosser-Property):

$$\begin{gathered} xx \Rightarrow x \hfill \\ x \Rightarrow xx \hfill \\ \end{gathered} $$

These rewrites are used to obtain a unique normal form for words in B and hence are the basis for a decision procedure for word equality in B.

The proof techniques are based uponterm rewriting systems [7] rather than the usual algebraic approach. Alternative and simpler proofs of a result reported earlier by Green and Rees [4] and Gerhardt [6] have been obtained.

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Communicated by K. Keimel

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Siekmann, J., Szabó, P. A Noetherian and confluent rewrite system for idempotent semigroups. Semigroup Forum 25, 83–110 (1982).

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  • Relative Length
  • Equational Theory
  • Unification Algorithm
  • Replacement Rule
  • Abstract Data Type