, Volume 29, Issue 3, pp 571-590

On the finite basis problem for certain 2-limited words

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Abstract

Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S(W) be the Rees quotient X*/I(W), where I(W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S(W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S(W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.

Supported by National Natural Science Foundation of China (Grant No. 10971086), Mathematical Tianyuan Foundation of China (Grant No. 11126186), Natural Science Foundation of Gansu Province (Grant No. 1107RJZA218) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2012-12)