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Finitely based words. (English) Zbl 1010.20043

Summary: Let \(W\) be a finite language and let \(W^c\) be the closure of \(W\) under taking subwords. Let \(S(W)\) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in \(W^c\). We call \(W\) finitely based if the monoid \(S(W)\) is finitely based. Although these semigroups have easy structure they behave “generically” with respect to the finite basis property [M. Jackson and O. Sapir, ibid. 10, No. 6, 683-708 (2000; Zbl 0980.20052)]. We describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

MSC:

20M07 Varieties and pseudovarieties of semigroups
68R15 Combinatorics on words
20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 0980.20052
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References:

[1] DOI: 10.1142/S0218196796000040 · Zbl 0844.08011 · doi:10.1142/S0218196796000040
[2] DOI: 10.1016/0021-8693(69)90058-1 · Zbl 0186.03401 · doi:10.1016/0021-8693(69)90058-1
[3] DOI: 10.1070/IM1988v030n02ABEH001012 · Zbl 0646.20047 · doi:10.1070/IM1988v030n02ABEH001012
[4] DOI: 10.1070/SM1988v061n01ABEH003199 · Zbl 0655.20045 · doi:10.1070/SM1988v061n01ABEH003199
[5] Shevrin L. N., VUZ) 29 (11) pp 1– (1985)
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