Two-parameter semigroups (Q2491183)

With \(S\) a topological semigroup with identity \(1\) and null element \(0\) \((\neq 1)\) containing two subsemigroups \(X\) and \(Y\) satisfying: (i) \(S= XY\), (ii) \(X\) and \(Y\) are continuous homomorphic images of the multiplicative unit interval \([0,1]\), (iii) \(X\cap Y= \{0,1\}\), the authors prove in Section 2 the fact that \(S\) has a unique factorization locally and that there exists a closed proper ideal \(I_0\) of \(S\) such that each member of \(S\setminus I_0\) can be written uniquely as \(xy\) and as \(y'x'\) for \(x\), \(x'\in X\) and \(y,y'\in Y\); furthermore, \(S\) is cancellative of \(I_0\) in the sense that if \(ss_1= ss_2\not\in I_0\) or \(s_1s= s_2s\not\in I_0\), then \(s_1= s_2\). In Section 3, the authors talk of `a ray' as the image of an injective one-parameter semigroup; a two parameter semigroup is a two-dimensional semigroup generated by two rays; the rectangle semigroup, the triangle semigroup and the wide-angle semigroup are also described in this section and it is proved that each of the four semigroups, the Abelian, the rectangle, the triangle and the wide-angle, is freely locally generated. With \(S\) as in Section 2, it is shown in Section 5 that either \(S\) or \(S\setminus\{0\}\) is the continuous homeomorphic image of the Abelian, triangle or wide-angle two-parameter semigroup, and this homomorphism is injective on some neighbourhood of the identity. In Section 6, with \(A\) a topological semigroup with identity \(1\), \(X\) and \(Y\) distinct continuous homeomorphic images of the additive semigroup \([0,\infty)\), both starting at \(1\) such that \(\overline X\) and \(\overline Y\) are compact and such that neither is contained in the group of units of \(A\), the authors show, that if \(A\) is uniquely divisible, then \(XY\) is isomorphic to one of the four basic semigroups.