Vol 56, No 3 (2017)

We study structures of degrees of stronger algorithmic reducibilities inside the degrees of weaker algorithmic ones. Results in this area are reviewed for algorithmic reducibilities m-, 1-, tt-, wtt-, T-, e-, s-, Q-, and we formulate questions that are still not settled for these. A computably enumerable Q-degree which consists of one computably enumerable m-degree is constructed.

We describe algebras of constants of the set of all partial derivations in free algebras of unitarily closed varieties over a field of characteristic 0. These constants are also called proper polynomials. It is proved that a subalgebra of proper polynomials coincides with the subalgebra generated by values of commutators and Umirbaev–Shestakov primitive elements p_m,n on a set of generators for a free algebra. The space of primitive elements is a linear algebraic system over a signature Σ = {[x, y], p_m,n | m, n ≥ 1}. We point out bases of operations of the set Σ in the classes of all algebras, all commutative algebras, right alternative and Jordan algebras.

A characterization of lattices with replaceable irredundant decompositions is given in the following six classes: the class of upper and lower continuous lattices; the class of upper continuous completely join-semidistributive lattices; the class of upper semimodular lower continuous lattices; the class of upper semimodular completely joinsemidistributive lattices; the class of consistent lower continuous lattices; the class of consistent completely join-semidistributive lattices.