After reviewing some notions of the formal theory of diierential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the eeects of local solvability and of gauge symmetries, respectively. We consider non-classical symmetry reductions and more general reductions using diierential constraints.

Theorem (1) There is an involution C of G satisfying: C(h) = h −1 (h ∈ H); (2) C(g) ∼ g −1 for all semisimple elements g; (3) Any two such involutions are conjugate by an inner automorphism;

Let S be a set of n ideals of a commutative ring A and let Geven (respectively Godd) denote the product of all the sums of even (respectively odd) number of ideals of S. If n ≤ 6 the product of Geven and the intersection of all ideals of S is included in Godd. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.

The concept of a 0-distributive poset is introduced. It is shown that a section semicomplemented poset is distributive if and only if it is 0-distributive. It is also proved that every pseudocomplemented poset is 0-distributive. Further, 0-distributive posets are characterized in terms of their ideal lattices.

It has been recently proved that a variety of associative PI-superalgebras with graded involution finite basic rank over field characteristic zero is minimal fixed ⁎-graded exponent if, and only it generated by subalgebra an upper block triangular matrix algebra, A:=UTZ2⁎(A1,…,Am), equipped suitable elementary Z2-grading involution. Here we give necessary sufficient conditions so IdZ2⁎(A) facto...