On the structure of (m,n)-commutative semigroups.
Let be an abelian semigroup, and a finite subset of . The sumset consists of all sums of elements of , with repetitions allowed. Let denote the cardinality of . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial such that for all sufficiently large . Lattice point counting is also used to prove that sumsets of the form have multivariate polynomial growth.
Generalized hypersubstitutions are mappings from the set of all fundamental operations into the set of all terms of the same language do not necessarily preserve the arities. Strong hyperidentities are identities which are closed under the generalized hypersubstitutions and a strongly solid variety is a variety which every its identity is a strong hyperidentity. In this paper we give an example of pre-strongly solid varieties of commutative semigroups and determine the least and the greatest pre-strongly...