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Complexity of hypersubstitutions and lattices of varieties

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Changphas T., Denecke K.

Discussiones Mathematicae - General Algebra and Applications

2003

tom 23

nr 1

31-43

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.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Changphas T.

1 Denecke K.

1 2003

Wyniki wyszukiwania

Complexity of hypersubstitutions and lattices of varieties