Pseudocharacters on semidirect products of semigroups (Q1326041)

A pseudocharacter on a semigroup \(S\) is a real-valued function \(f\) on \(S\) satisfying the following conditions: 1) The set \(\{f(xy) - f(x) - f(y)\); \(x, y \in S\}\) is bounded; 2) \(f(x^ n) = nf(x)\) for any \(x \in S\) and any natural number \(n\). The set of all the pseudocharacters of a semigroup \(S\) is a real-valued vector space with respect to the usual operations and it is denoted by \(PX(S)\). If \(S\) is an arbitrary semigroup, and \(S^ 1\) is the semigroup obtained from \(S\) by adjunction of a unit, then it is easy to see that \(PX(S) = PX(S^ 1)\). Let \(A\) and \(B\) be semigroups with units and let the semigroup \(G\) be the semidirect product of its subsemigroups \(A\) and \(B\), where the semigroup \(A\) acts on the semigroup \(B\). The pseudocharacter \(\varphi\) of the semigroup \(B\) is invariant with respect to the action of \(A\) if for any \(h \in A\) the equality \(\varphi(g^ h) = \varphi(g)\) is satisfied. The subspace of \(PX(B)\) consisting of all \(A\)-invariant pseudocharacters of \(B\) is denoted by \(PX(B,A)\). Theorem. \(PX(G) = PX(A) + PX(B,A)\).