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The breadth of the lattice of those varieties of inverse semigroups which contain the variety of groups

  • Norman R. Reilly (a1)

Synopsis

The relations v1 and v2 defined on the lattice ℒ of varieties of inverse semigroups by v1 if and only if and v2 if and only if , where denotes tie variety of groups, are both congruences on ℒ the class v1, is simply the lattice of varieties of grcups and is therefore known to have cardinality .

The class v2 is precisely the sublattice of consisting of those varieties containing . Each v1-class contains preciselyone element of v2. The main result of this paper establishes that the sublattice v2 of has breadth . From this it follows that the lattice ℒ/v1 also has breadth . Some consequences concerning varieties generated by fundamental inverse semigroups are also considered.

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1Adyan, S.. The Bumside problem and identities in groups. Ergebnisse der Mathematik (Berlin: Springer, 1979).
2Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).
3Kleiman, E. I.. On the lattice of varieties of inverse semigroups. Izv. Vysš. Učeb. Zabend. Matematika 7 (1976), 106109.
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5Munn, W. D.. Fundamental inverse semigroups. Quart. J. Math. Oxford 21 (1970), 157170.
6Olshansky, A. U.. On the problem of finite basis of identities in groups. Izv. Akad.Nauk SSSR Ser. Mat. 34 (1970), 176184. (In Russian).
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The breadth of the lattice of those varieties of inverse semigroups which contain the variety of groups

  • Norman R. Reilly (a1)

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