POINTWISE AND GLOBAL SUMS AND NEGATIVES OF TRANSLATION RELATIONS
Tamás Glavosits, Árpád Száz · 2004
ArelationFon a groupoid X to a set Y is called a pointwise translation relation if F (y)⊂F(x+y) forall x, y ∈ X. Moreover, arelationFon a groupoid X is called a global translation relation if x + F (y) ⊂ F ( x + y) for all x, y ∈ X. After establishing some basic properties of the translation relations, we prove some simple theorems about the pointwise and global sums and negatives of translation relations. For instance, we show that if F is a normal and G is an arbitrary global translation relation on a group, then the global sum of F and G coincides with the composition of G and F. And ˙ the global negative of F coincides with the inverse of F. Global translation functions and relations play important roles in the extensions and uniformizations of semigroups and groups, respectively. While the pointwise ones seem to have no such applications despite that they are also closely related to additive relations.