Closed sets of finitary functions between products of finite fields of coprime order

We investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$ ∏ j = 1 m F q j = K to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$ ∏ i = 1 n F p i = F , where $$|{\mathbb K}|$$ | K | and $$|{\mathbb {F}}|$$ | F | are coprime. An $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$ F p [ K × ] -submodules of $$\mathbb {F}_p^{{\mathbb K}}$$ F p K , where $${\mathbb K}^{\times }$$ K × is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$ K = ∏ i = 1 m F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoids.

On the Grothendieck ring of varieties

Let K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

A THEOREM ON MATRIX GROUPS

Let K be a principal ideal domain, and An, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KAn\{0} and U(KAn) the group of units of KAn. For a multiplicative monoid P generated by U(KAn) and Ω, we prove that any generating set for [Formula: see text] contains infinitely many elements not in [Formula: see text]. Furthermore, we present a way of constructing elements of [Formula: see text] not in [Formula: see text] for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

Primary decompositions over domains

Throughout, R denotes a commutative domain with 1, and Q (≠R) its field of quotients, which is viewed here as an R-module. The symbol K will stand for the R-module Q/R, while R denotes the multiplicative monoid R/0.