Complexity of hypersubstitutions and lattices of varieties

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 ${\displaystyle M=H{₂}^{op}}$ 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 ${\displaystyle M=H{ₙ}^{op}}$ we determine the complete lattices of all M-solid varieties of semigroups.