We establish a unique factorization result into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] whose determinant is equal to 1, where K is a field, and where multiplication is defined as the usual matrix-multiplication if the degrees of the factors add up. This investigation is motivated by a result on matrices of entire functions.
A regular hypersubstitution is a mapping which takes every -ary operation symbol to an -ary term. A variety is called regular-solid if it contains all algebras derived by regular hypersubstitutions. We determine the greatest regular-solid variety of semigroups. This result will be used to give a new proof for the equational description of the greatest solid variety of semigroups. We show that every variety of semigroups which is finitely based by hyperidentities is also finitely based by identities....
We enlarge the problem of valuations of triads on so called lines. A line in an -structure (it means that is a semigroup and is an automorphism or an antiautomorphism on such that ) is, generally, a sequence , , (where is the class of finite integers) of substructures of such that holds for each . We denote this line as and we say that a mapping is a valuation of the line in a line if it is, for each , a valuation of the triad in . Some theorems on an existence of...