Green's \(\mathcal R\), \(\mathcal D\) and \(\mathcal H\) relations for generalized polynomials (Q1375878)

Let \(S(\mathbb{R})\) denote the semigroup of all continuous functions of the set \(\mathbb{R}\) of all reals into itself. A function \(f\in S(\mathbb{R})\) is called a generalized polynomial if \[ \lim_{x\to\pm\infty}f(x)=\pm\infty \] and if there exists a finite set \(\{a_i\}_{i=0}^n\) of reals such that \(f\) is injective on intervals \((-\infty,a_0\rangle\), \(\langle a_n,+\infty)\) and \(\langle a_{i-1},a_i\rangle\) for all \(i=1,\dots,n\). We say that \(f\) is of even degree if \(\lim_{x\to-\infty}f(x)=\lim_{x\to+\infty}f(x)\) and of odd degree otherwise. If \(\lim_{x\to+\infty}=+\infty\) then \(f\) is positive and it is negative otherwise. Let \(GP(\mathbb{R})\) denote the set of all generalized polynomials, and \(GPO(\mathbb{R})\), or \(GPE(\mathbb{R})\) denote the set of all generalized polynomials of odd, or even respectively, degree. It is proved that \(f\in S(\mathbb{R})\) is a generalized polynomial if and only if \(f=g\circ h\) for a polynomial \(g\in S(\mathbb{R})\) and a bijective increasing homeomorphism \(h\in S(\mathbb{R})\). Further, \(GP(\mathbb{R})\) is a subsemigroup of \(S(\mathbb{R})\), \(GPE(\mathbb{R})\) is a two-sided ideal in \(GP(\mathbb{R})\), the collections of all positive \(f\in GPE(\mathbb{R})\) and all negative \(f\in GPE(\mathbb{R})\) are right ideals in \(GP(\mathbb{R})\), and \(S(\mathbb{R})\setminus GPO(\mathbb{R})\) is a prime ideal in \(S(\mathbb{R})\). For a Green's relation \(\mathcal G\) and for \(f,g\in GP(\mathbb{R})\) we have that \(f\) and \(g\) are \(\mathcal G\)-equivalent in \(GP(\mathbb{R})\) just when \(f\) and \(g\) are \(\mathcal G\)-equivalent in \(S(\mathbb{R})\). Every \(f\in GPE(\mathbb{R})\) is contained in infinitely many principal right ideals in \(S(\mathbb{R})\) but every \(f\in GPO(\mathbb{R})\) is contained in only finitely many principal right ideals in \(S(\mathbb{R})\) (an estimation is given). For any \(f\in GPO(\mathbb{R})\) there exist finite collections \(\{p_i\}_{i=1}^n\) and \(\{q_j\}_{j=1}^m\) of polynomials such that if \(f=g\circ h\) for \(g,h\in S(\mathbb{R})\) then \(g=p_i\circ u\), \(h=v\circ q_j\circ w\) for some \(i=1,\dots,n\), \(j=1,\dots,m\) and some bijective homeomorphisms \(u,v,w\in S(\mathbb{R})\). The Green's relation \(\mathcal R\) on \(GP(\mathbb{R})\) is fully described, the Green's relations \(\mathcal H\) and \(\mathcal D\) on \(GP(\mathbb{R})\) are partially described. The Green's relation \(\mathcal J\) on \(GPO(\mathbb{R})\) coincides with the Green's relation \(\mathcal D\) and all \(f,g\in GPE(\mathbb{R})\) are \(\mathcal J\)-equivalent in \(GP(\mathbb{R})\).