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Semigroup varieties closed for the Bruck extension. (English) Zbl 0813.20063

For any semigroup \(S\), let \(S^ 1 = S\), if \(S\) is a monoid, otherwise \(S^ 1\) denotes \(S\) with an identity adjoined. Let \(\mathbb{N}\) denote the set of natural numbers. The Bruck extension, \(B(S)\), of \(S\) is the semigroup with underlying set \(\mathbb{N} \times S^ 1 \times \mathbb{N}\) and operation defined by: \[ (m,s,n)(p,t,q) = \begin{cases} (m,s,q+n - p) & \text{if \(n > p\)};\\ (m,st,q) & \text{if \(n = p\)};\\ (m + p - n,t,q) & \text{if \(p > n\).}\end{cases} \] A subvariety \(\mathbf V\) of the variety \(\mathbf S\) of all semigroups is said to be closed for the Bruck extension if \(S \in {\mathbf V}\) implies \(B(S) \in {\mathbf V}\). For an arbitrary \(\mathbf V\), \(B({\mathbf V})\) denotes the least Bruck-closed variety containing \(\mathbf V\). In a previous paper [J. Algebra 163, 777-794 (1994; Zbl 0811.20052)] the authors showed that the lattice \(BL({\mathbf S})\) of Bruck-closed semigroup varieties is a complete sublattice of \(L({\mathbf S})\), the lattice of all semigroup varieties. The main result of this paper shows that \(BL({\mathbf S})\) contains a chain of order type that of the real numbers. Other results show that if \(\mathbf V\) is a proper subvariety of \(\mathbf S\) then \(B({\mathbf V})\) is a proper subvariety of \(\mathbf S\) and that any proper Bruck-closed variety is the least element of an infinite chain in \(BL({\mathbf S})\).

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties

Citations:

Zbl 0811.20052
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References:

[1] Rosenstein, Linear orderings (1982)
[2] Petrich, Inverse semigroups (1984)
[3] DOI: 10.1006/jabr.1994.1043 · Zbl 0811.20052 · doi:10.1006/jabr.1994.1043
[4] Adjan, Dokl. Akad. Nauk SSSR 43 pp 499– (1962)
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[7] DOI: 10.1007/BF01196094 · Zbl 0792.20055 · doi:10.1007/BF01196094
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