Geodesic
Complexity of hypersubstitutions and lattices of varieties
Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 31-43

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Hypersubstitutions are mappings which map operation symbols to terms. The set of all hypersubstitutions of a given type forms a monoid with respect to the composition of operations. Together with a second binary operation, to be written as addition, the set of all hypersubstitutions of a given type forms a left-seminearring. Monoids and left-seminearrings of hypersubstitutions can be used to describe complete sublattices of the lattice of all varieties of algebras of a given type. The complexity of a hypersubstitution can be measured by the complexity of the resulting terms. We prove that the set of all hypersubstitutions with a complexity greater than a given natural number forms a sub-left-seminearring of the left-seminearring of all hypersubstitutions of the considered type. Next we look to a special complexity measure, the operation symbol count op(t) of a term t and determine the greatest M-solid variety of semigroups where M = H₂^op is the left-seminearring of all hypersubstitutions for which the number of operation symbols occurring in the resulting term is greater than or equal to 2. For every n ≥ 1 and for M = Hₙ^op we determine the complete lattices of all M-solid varieties of semigroups.
Keywords: hypersubstitution, left-seminearring, complexity ofa hypersubstitution, M-solid variety 
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Changphas, Thawhat; Denecke, Klaus. Complexity of hypersubstitutions and lattices of varieties. Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 31-43. http://geodesic.mathdoc.fr/item/DMGAA_2003_23_1_a3/