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Gelfand-Kirillov dimension of some primitive abundant semigroups. (English) Zbl 1310.20047

Summary: The growth and Gelfand-Kirillov dimension of some primitive abundant semigroups are investigated. It is shown that for certain primitive abundant (regular) semigroup \(S\), \(S\) as well as the semigroup algebra \(K[S]\) has polynomial growth if and only if all of its cancellative submonoids (subgroups) \(T\) as well as \(K[T]\) have polynomial growth. As applications, it is shown that if \(S\) is a finitely generated primitive inverse monoid having the permutational property, then \(\mathrm{clK}\dim K[S]=\mathrm{GK}\dim K[S]=\mathrm{rk}(S)\).

MSC:

20M05 Free semigroups, generators and relations, word problems
16P90 Growth rate, Gelfand-Kirillov dimension
20M25 Semigroup rings, multiplicative semigroups of rings
16S36 Ordinary and skew polynomial rings and semigroup rings
20M17 Regular semigroups
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