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On some finiteness conditions in semigroup and group theory. (English) Zbl 0808.20046

The paper answers in the negative questions stated by L. N. Shevrin [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 553-566 (1965; Zbl 0133.279)].
A semigroup \(S\) satisfies the maximal (minimal) condition – shortly is \((Mx)\) (\((Mn)\)) – if every ascending (descending, respectively) chain stabilizes. A semigroup \(S\) has finite rank \(r\) (shortly is \((Rn)\)) if it has a generating set of \(r\) elements and \(r\) is the least such number. A semigroup \(S\) is of finite breadth \(b\) (shortly is \((Br)\)) if for every set of \(m\) elements \(m > b\), \(S\) has a subset with \(b\) elements both generating the same subsemigroup, and \(b\) is the least such number. A semigroup \(S\) has finite length \(\ell\) (is \((Ln)\)) if in every ascending chain of subsemigroups of \(S\) consisting of \(\ell + 2\) terms there is a repetition and \(\ell\) is the least such number.
Obviously condition \((Ln)\) implies conditions \((Mx)\), \((Mn)\) and \((Br)\), moreover \((Br)\) implies \((Rn)\). It is proved that (a) \((Mx)\) and \((Mn)\) and \((Br)\) do not imply \((Ln)\), (b) \((Mx)\) and \((Mn)\) and \((Rn)\) do not imply \((Br)\), (c) \((Mx)\) and \((Mn)\) do not imply \((Rn)\), (d) \((Mx)\) and \((Br)\) do not imply \((Mn)\), (e) \((Mn)\) and \((Br)\) do not imply \((Mx)\).

MSC:

20M05 Free semigroups, generators and relations, word problems
20E15 Chains and lattices of subgroups, subnormal subgroups

Citations:

Zbl 0133.279
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