ACTA issues

## Semigroup operations which are distributive over a given semigroup operation on positive real numbers

Hirokazu Oka, Takeshi Miura, Sin-Ei Takahasi

Acta Sci. Math. (Szeged) 86:3-4(2020), 493-502
116/2020

 Abstract. Let $\mathbf R_+$ be the space of positive real numbers with the ordinary topology and let $\star$ be an arbitrary cancellative continuous semigroup operation on $\mathbf R_+$ or some special noncancellative continuous semigroup operation on $\mathbf R_+$. We characterize the set $\mathcal {D}_{\star }^{-1}(\mathbf R_+)$ of all cancellative continuous semigroup operations on $\mathbf R_+$ which are distributive over $\star$ in terms of homeomorphism. As a consequence, it is shown that if $\star$ is homeomorphically isomorphic to the ordinary addition $+$ on $\mathbf R_+$, any element of $\mathcal {D}_{\star }^{-1}(\mathbf R_+)$ is homeomorphically isomorphic to the ordinary multiplication on $\mathbf R_+$, and that if $\star$ is cancellative and not homeomorphically isomorphic to $+$, then $\mathcal {D}_{\star }^{-1}(\mathbf R_+)$ is empty. Moreover, if $\star$ is homeomorphically isomorphic to some special noncancellative continuous semigroup operation on $\mathbf R_+$, $\mathcal {D}_{\star }^{-1}(\mathbf R_+)$ is also shown to be empty. DOI: 10.14232/actasm-020-116-1 AMS Subject Classification (1991): 22A15; 06F05 Keyword(s): cancellative semigroup operation, distributive law, homeomorphic isomorphism, noncancellative semigroup operation received 16.1.2020, revised 16.8.2020, accepted 21.8.2020. (Registered under 116/2020.)