Operators on the Lattice of Pseudovarieties of Finite Semigroups
- Cite this article as:
- Reilly, N. & Zhang, S. Semigroup Forum (1998) 57: 208. doi:10.1007/PL00005974
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(F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [Uρ, Uρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of Uρ, bases of pseudoidentities for Uρ, instances of commutativity (Uρ)σ = Uσ)ρ, as well as the semigroups generated by certain pairs of such operators.