## Finitely based, finite sets of words

Venue: | Internat. J. Algebra Comput |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Jackson_finitelybased,,

author = {Marcel Jackson and Olga Sapir},

title = {Finitely based, finite sets of words},

journal = {Internat. J. Algebra Comput},

year = {},

volume = {10},

pages = {683--708}

}

### OpenURL

### Abstract

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U ⊃ W and V ⊃ W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based (regardless of the situation for S(W)). The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example. 1

### Citations

35 | Algorithmic problems in varieties
- Kharlampovich, Sapir
- 1995
(Show Context)
Citation Context ... containing it is also NFB. There exist semigroups with every consistent combination of these properties: FB; INFB; and NFB but not INFB. The term “weakly finitely based” (WFB) has been introduced in =-=[1]-=- to denote those locally finite algebras that are not INFB. Likewise a semigroup will be called “weakly not finitely based” (WNFB) if it is NFB but not INFB (that is, from the intersection of the clas... |

21 |
Tarski’s finite basis problem is undecidable
- McKenzie
- 1996
(Show Context)
Citation Context ...FB (that is, from the intersection of the class of WFB semigroups and the class of NFB semigroups). The classes of FB and INFB finite algebras were shown to be recursively inseparable by R. McKenzie (=-=[2]-=-), giving a negative solution to one of the most famous problems in the study of varieties, Tarski’s Finite Basis Problem. For semigroups there is a very large volume of work devoted to investigating ... |

11 |
Inherently non-finitely based finite semigroups
- Sapir
- 1987
(Show Context)
Citation Context ...to investigating the finite basis problem (see [10] for example) and in contrast with McKenzie’s result, a powerful description of the INFB finite semigroups has been obtained by M. V. Sapir ([6] and =-=[7]-=-). Algorithmically classifying the classes of FB and WNFB semigroups still remains a difficult and unsolved problem. In this paper we investigate an interesting class of finite aperiodic semigroups (t... |

9 |
Problems of Burnside type and the finite basis property in varieties of semigroups
- Sapir
- 1988
(Show Context)
Citation Context ...devoted to investigating the finite basis problem (see [10] for example) and in contrast with McKenzie’s result, a powerful description of the INFB finite semigroups has been obtained by M. V. Sapir (=-=[6]-=- and [7]). Algorithmically classifying the classes of FB and WNFB semigroups still remains a difficult and unsolved problem. In this paper we investigate an interesting class of finite aperiodic semig... |

8 | Basis questions for general algebras, Algebra Universalis 19 - Perkins - 1984 |

6 |
On Cross semigroup varieties and related questions
- Sapir
- 1990
(Show Context)
Citation Context ...act they are FB anyway). The Theorem now follows since by Example 2, S({abab}) is NFB. 5 Joins of Varieties Generated by Monoids of the Form S(W ) Examples found by M. Volkov (see [10]) and M. Sapir (=-=[8]-=-) show that the class of finite FB semigroups and the class of finite NFB semigroups are not closed under taking subsemigroups, homomorphic images, or direct products. The presence of nontrivial subgr... |

2 |
Bases for equational theories of semigroups, J.Algebra11
- Perkins
- 1969
(Show Context)
Citation Context ...e word w then we will say that w is a FB (or NFB) word if W = {w} is a FB (or NFB, respectively) set of words. 1sThe identities of semigroups of the form S(W ) have been of interest since P. Perkins (=-=[3]-=-) showed that S({abcba, acbab, abab, aab}) was NFB, one of the first examples of a finite NFB semigroup. It is clear from the results in [6] and [7] however that a semigroup S(W ) is never INFB. This ... |

2 |
Finitely based words
- Sapir
(Show Context)
Citation Context ...Question 7.1 of [10]) is the following: Question 1.3 Which finite sets of words are FB? A partial solution to Question 1.3 has been obtained by the second author of this paper: Theorem 1.4 (O. Sapir, =-=[9]-=-) If w is an element of {a, b} ∗ then w is a FB word if and only if it is one of the following words: a n b m , b n a m , a n ba m , or b n ab m for some n and m. This shows that “most” words in a two... |

1 |
On almost simple semigroup identities
- Pollak, Volkov
- 1985
(Show Context)
Citation Context ...our letter words w involving two or less distinct letters for which S({w}) has less than 10 elements are (up to a change in letter names) aaaa and abab. The word aaaa has only 4 distinct subwords. In =-=[5]-=- it is shown that if a semigroup satisfies xyx ≈ xxy or xyx ≈ yxx then it is FB. If xxxx is an isoterm then in order that S(W ) not satisfy one of these identities, either xyx or both xyy and yyx must... |