By using several auxiliary results on relations and their intersection convolutions, we give some necessary and sufficient conditions in order that a certain additive partial selection relation Φ of a relation F of one group X to another Y could be extended to a total, additive selection relation Ψ of the relation F + Φ(0).The results obtained extend some Hahn-Banach type extension theorems of B. Rodríguez-Salinas, L. Bou, Z. Gajda, A. Smajdor, W. Smajdor, and the second author. Moreover, they can be used to prove some alternate forms of the Hyers-Ulam type selection theorems of Z. Gajda, R. Ger, R. Badora, Zs. Páles, and the second author.Keywords Additive and homogeneous relations · Intersection convolutions of relations · Extensions of additive partial selection relations

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The most important particular cases of the main results of this paper say that: A subset A of the group X has a nonnegativity domain if and only if A is symmetric and 2-cancellable. Moreover, A has an additive nonnegativity domain if and only if A is symmetric and perfectly cancellable. In addition, it is shown that a commutative subset of X is perfectly cancellable if and only if it is infinitely cancellable.Here, the set A is called n-cancellable for some natural n if nx = 0 implies x = 0 for all x € A. In particular, A is called infinitely cancellable if it is n-cancellable for all n. Moreover, A is called perfectly cancellable if for any additive antisymmetric subset B of A and any x € A \ (B U (-B)) we have either 0 ^ Fg(x) or 0 ^ Fg (-x), where Fg(x) is the additive hull of (B U {a:}) \ {0}.The results obtained are illustrated here only with the help of additive and multiplicative groups of complex numbers. Their applications to odd, additive and translation relations are postponed to subsequent papers. We do not compare our results with the more complicated results of P. Conrad, R. Botto Mura and A. Rhemtulla on the extensions of partial right-orders to total ones in multiplicative groups. 1991 Mathematics Subject Classification: 06P15. A few basic facts on families of sets and subsets of groups

We study some new logarithmic functional equations and their Pexiderizations on different structures.

We show that the sum of two intervals in an ordered dense Abelian group is also an interval such that the endpoints of the sum are equal to the sums of the endpoints. We prove analogous statements concerning to the product of two intervals.