The operators of rotoids in homogeneous spaces or generalized ordered spaces

A space X is a rotoid if there are a special point e ∈ X and a homeomorphism H from X 2 onto itself such that H ( x , x ) = ( x , e ) and H ( e , x ) = ( e , x ) for each x ∈ X . Rotoids are generalizations of topological groups, and the Sorgenfrey line is a rotoid and not a topological group. In this paper, we prove some cardinal invariants for rotoids, dichotomy theorems in generalized ordered spaces that are rotoids and dichotomy theorems for remainders in Hausdorff compactifications of paratopological groups that are rotoids. We show, for example, that χ ( X ) = π χ ( X ) holds in the class of strong rotoids, that any snf rectifiable space is sof and that any GO-space which is a rotoid is hereditarily paracompact giving an affirmative answer to H. Bennett, D. Burke and D. Lutzerʼs question [11]. We also show that any homogeneous GO-space which is a rotoid is first-countable or a totally disconnected P-space, that any remainder of a Hausdorff compactification of a paratopological group which is a rotoid is either pseudocompact or Lindelöf. Moreover, some open questions concerning rotoids are posed.